Samacheer Class 12 Physics - Electrostatics

According to the principle of quantisation of charges, the net charge ($q$) of any object in the universe can only be an integral multiple of the fundamental unit of electric charge ($e$).

Where:

• $q$ is the total charge of the body.
• $n$ is any integer ($\dots, -3, -2, -1, 0, 1, 2, 3, \dots$).
• $e$ is the fundamental unit of charge (the charge of an electron or proton), which has a magnitude of approximately $1.6 \times 10^{-19}\text{ C}$.

Coulomb's law states that the electrostatic force between two point charges at rest is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

Representation of terms:

• $\vec{F}_{21}$ : Electrostatic force exerted on charge $q_2$ by charge $q_1$.
• $q_1, q_2$ : Magnitudes of the two interacting point charges.
• $r_{12}$ : Center-to-center distance between the charges $q_1$ and $q_2$.
• $\hat{r}_{12}$ : Unit vector directed from charge $q_1$ to charge $q_2$.
• $\varepsilon_0$ : Permittivity of free space (vacuum), valued at approximately $8.854 \times 10^{-12}\text{ C}^2\text{ N}^{-1}\text{ m}^{-2}$.
• $\frac{1}{4\pi\varepsilon_0}$ : Proportionality constant, which equals $9 \times 10^9\text{ N m}^2\text{ C}^{-2}$ in vacuum or air.

| Property | Coulomb Force | Gravitational Force |

| --- | --- | --- |

| Nature of Force | Can be either attractive or repulsive depending on the nature of the interacting charges. | It is always purely attractive in nature between masses. |

| Dependence on Medium | It strongly depends on the nature of the intervening medium between the charges ($\varepsilon$). | It is entirely independent of the medium surrounding the masses. |

| Magnitude Comparison | It is much stronger in magnitude (e.g., the electrostatic force between two protons is roughly $10^{36}$ times stronger than their gravitational attraction). | It is a comparatively weak natural force. |

| Proportionality Constant | The constant factor $k = \frac{1}{4\pi\varepsilon_0} = 9 \times 10^9\text{ N m}^2\text{ C}^{-2}$ is exceptionally large. | The universal gravitational constant $G = 6.67 \times 10^{-11}\text{ N m}^2\text{ kg}^{-2}$ is exceptionally small. |

Coulomb's law only explains the interaction between two point charges at a time. When multiple charges interact simultaneously, the Superposition Principle is used.

It states that the total electrostatic force exerted on a single specific point charge by a collection of surrounding point charges is equal to the vector sum of the individual electrostatic forces exerted on it by each of the other charges, calculated as if the other charges were acting independently.

Where $\vec{F}_{1}^{\text{total}}$ is the net force acting on charge $q_1$, and $\vec{F}_{12}, \vec{F}_{13}$ represent the independent forces exerted on $q_1$ by charges $q_2, q_3$ respectively.

The electric field at a point in space is defined as the electrostatic force experienced per unit positive test charge placed at that point. It describes the region of influence surrounding a charge distribution.

Where:

• $\vec{E}$ is the electric field vector.
• $\vec{F}$ is the electrostatic force experienced by the test charge.
• $q_0$ is the magnitude of the positive test charge.

Key Attributes:

• It is a vector quantity.
• Its SI unit is Newton per Coulomb ($\text{N C}^{-1}$) or Volt per meter ($\text{V m}^{-1}$).

Electric field lines are a set of continuous, imaginary curves or paths drawn in space to visually represent the magnitude and direction of an electric field within a region.

Key Characteristics:

• The tangent drawn to an electric field line at any given point gives the direction of the electric field vector ($\vec{E}$) at that point.
• They begin on positive charges and terminate on negative charges.
• The relative density (closeness) of the lines indicates the strength of the electric field; closer lines represent a strong electric field, while wider spaced lines represent a weak field.

If two electric field lines were to intersect or cross each other at a point, it would be possible to draw two distinct tangents at that single intersection point.

Because a tangent indicates the direction of the electric field, this would imply that the electric field vector points in two different directions simultaneously at the exact same location. Physically, the net electric field at any specific point must be unique and have only one resultant direction. Therefore, electric field lines can never intersect.

An electric dipole consists of a system of two equal and opposite point charges ($+q$ and $-q$) separated by a small, fixed distance ($2a$).

Magnitude of Dipole Moment:

The magnitude of the electric dipole moment ($p$) is equal to the product of the magnitude of one of the charges ($q$) and the separation distance ($2a$) between them:

Direction:

By physics convention, the electric dipole moment is a vector quantity ($\vec{p}$) directed along the axis of the dipole, pointing from the negative charge ($-q$) to the positive charge ($+q$).

For an arbitrary distribution or collection of $n$ discrete point charges, the general electric dipole moment ($\vec{p}$) is defined mathematically as the vector sum of the product of each individual charge with its respective position vector relative to a chosen origin.

Where:

• $q_i$ is the magnitude and sign of the $i$-th point charge.
• $\vec{r}_i$ is the position vector of the $i$-th charge relative to the origin of the coordinate system.

The electrostatic potential ($V$) at any given point in an electric field is defined as the amount of work done by an external agent in moving a unit positive test charge with a constant velocity (no acceleration) from infinity to that specific point against the electrostatic forces of the field.

Key Attributes:

• It is a scalar quantity.
• Its SI unit is the Volt ($\text{V}$) or Joule per Coulomb ($\text{J C}^{-1}$).

An equipotential surface is any continuous surface over which the electrostatic potential ($V$) remains completely uniform and identical at every single point.

Because the potential is the same everywhere on the surface ($V_A = V_B$), the potential difference ($\Delta V$) between any two points lying on an equipotential surface is exactly zero. As a result, no work is required to move any electric charge along an equipotential surface.

The primary properties of an equipotential surface include:

1. Zero Work Done: The work done in moving a charge $q$ between any two points $A$ and $B$ on an equipotential surface is zero ($W = q(V_B - V_A) = 0$).

2. Perpendicular Electric Field: The electric field ($\vec{E}$) is always oriented normally (perpendicularly) to the equipotential surface at every point. If it weren't perpendicular, there would be a tangential field component along the surface, which would require work to move a charge.

3. No Intersection: Two different equipotential surfaces can never intersect each other. If they did, there would be two different values of potential at the line of intersection, which is physically impossible.

The electric field ($\vec{E}$) and the electric potential ($V$) are related through the concept of a spatial gradient. The electric field at any point is equal to the negative gradient of the electric potential at that point.

The negative sign indicates that the direction of the electric field vector always points toward the direction where the electric potential decreases most rapidly.

The electrostatic potential energy ($U$) of a system of point charges is defined as the total amount of work done by an external agent in assembling the charges by bringing them from an initial separation of infinity to their respective current positions in the configuration, without any acceleration.

For a simple two-charge system ($q_1$ and $q_2$) separated by a distance $r$, it is given by:

Electric flux ($\Phi_E$) is defined as the total number of electric field lines passing normally (perpendicularly) through a given surface area. It provides a measure of the throughput of an electric field across a surface.

Where:

• $\vec{E}$ is the electric field vector.
• $d\vec{A}$ is the area vector (perpendicular to the surface element).

Key Attributes:

• It is a scalar quantity.
• Its SI unit is $\text{N m}^2\text{ C}^{-1}$ or $\text{V m}$.

Electrostatic energy density ($u_E$) is defined as the total electrostatic potential energy stored per unit volume in the electric field between charges or plates.

Where:

• $\varepsilon_0$ is the permittivity of free space.
• $E$ is the magnitude of the electric field in that region.
• Its SI unit is Joules per cubic meter ($\text{J m}^{-3}$).

Electrostatic shielding is the process of isolating a specific region of space from the effects of external electric fields.

This phenomenon is based on the property that the electric field inside a hollow conductor (cavity) is completely zero, regardless of the magnitude of charges present outside or on the surface of the conductor. Any external electric field causes a redistribution of charges on the outer surface of the conductor in such a way that the field lines terminate on the exterior, leaving the interior cavity entirely unaffected.

Practical Application:

During a thunderstorm accompanied by lightning, it is much safer to remain inside a car than to stand under a tree or in the open. The metallic body of the car acts as an electrostatic shield, ensuring that the electric field inside remains zero and any lightning strike passes safely over the outer surface to the ground.

Polarisation ($\vec{P}$) is defined as the total induced dipole moment per unit volume of a dielectric material when it is placed inside an external electric field.

When an external electric field ($\vec{E}_0$) is applied to a non-polar or polar dielectric material, the positive and negative charges experience forces in opposite directions. This induces tiny electric dipole moments throughout the material, aligning them along the direction of the applied field.

For a linear isotropic dielectric, the polarisation is directly proportional to the external electric field:

Where $\chi_e$ is a dimensionless constant known as the electric susceptibility of the dielectric medium.

Dielectric strength is defined as the maximum external electric field that a dielectric material can withstand before its chemical bonds break down and it begins to conduct electricity.

When the external electric field applied across a dielectric exceeds this threshold value, the field becomes strong enough to strip electrons away from their parent atoms. This transforms the insulating material into a conductor, a phenomenon known as dielectric breakdown.

For example, the dielectric strength of air is approximately $3 \times 10^6\text{ V m}^{-1}$. If the electric field in air exceeds this value, air breaks down and becomes conducting, which is observed as a spark or lightning.

The capacitance ($C$) of a conductor or a capacitor is defined as the ratio of the magnitude of charge ($Q$) given to it to the resulting change in its electric potential ($V$).

It represents the ability of a system to store electric charge and electrical potential energy.

Unit:

• The SI unit of capacitance is the Farad ($\text{F}$), named in honor of Michael Faraday.
• Since $C = \frac{Q}{V}$, $1\text{ Farad} = 1\text{ Coulomb per Volt } (\text{C V}^{-1})$.
• In practical circuits, much smaller sub-multiples are used, such as the microfarad ($1\mu\text{F} = 10^{-6}\text{ F}$) and picofarad ($1\text{pF} = 10^{-12}\text{ F}$).

Corona discharge, also known as the action of points, is the phenomenon where the electric charge accumulated at the sharp pointed ends of a conductor leaks out into the surrounding air due to high localized electric fields.

Mechanism:

1. According to the distribution of charges, the surface charge density ($\sigma = \frac{Q}{A}$) is inversely proportional to the radius of curvature. At sharp points or edges, the radius is extremely small, making the surface charge density exceptionally high.

2. This creates an extremely strong localized electric field near the sharp point, which is high enough to ionize the surrounding air molecules.

3. The positive ions produced during ionization are repelled by the positive sharp point, while the negative ions are attracted to it, partially neutralizing the charge on the point. This movement of ions sets up an "electric wind" and results in a continuous leakage of charge.

There are three fundamental properties of electric charges that govern all electrostatic phenomena:

#### 1. Electric Charge

Electric charge is an intrinsic property of the fundamental particles (like electrons and protons) that make up matter. It is responsible for the electrostatic forces between them.

• There are two types of electric charges: positive and negative. Like charges repel each other, and unlike charges attract each other.
• The SI unit of electric charge is the Coulomb ($\text{C}$).

#### 2. Conservation of Charges

According to the law of conservation of electric charge, the total electric charge in an isolated system remains constant over time. Charge can neither be created nor destroyed; it can only be transferred from one body to another.

• For example, when a glass rod is rubbed with a silk cloth, electrons are transferred from the glass rod to the silk cloth. The glass rod gains a positive charge, and the silk cloth acquires an equal amount of negative charge. The net total charge of the system (glass rod + silk) remains zero both before and after rubbing.

#### 3. Quantisation of Charges

The charge ($q$) of any object is not continuously variable but is restricted to discrete values. It must be an integral multiple of the fundamental unit of charge ($e$).

Where:

• $n$ is an integer ($n = 0, \pm 1, \pm 2, \pm 3, \dots$).
• $e$ is the charge of an electron or proton, which has a magnitude of approximately $1.6 \times 10^{-19}\text{ C}$.
• At the macroscopic level, the discrete nature of charge is ignored because the total charge involved is usually vast compared to $e$, making the charge distribution appear continuous.

Coulomb’s Law describes the quantitative electrostatic force between two stationary point charges.

#### Statement

The electrostatic force between two point charges at rest is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The force acts along the straight line joining the two charges.

In vector form, the force $\vec{F}_{21}$ exerted on charge $q_2$ by charge $q_1$ separated by a distance $r$ is:

Where $\hat{r}_{12}$ is the unit vector pointing from $q_1$ to $q_2$, and $\varepsilon_0$ is the permittivity of free space.

#### Various Aspects of Coulomb's Law

1. Inverse Square Dependence: The force decreases inversely with the square of the distance ($F \propto \frac{1}{r^2}$). If the distance between two charges is doubled, the force between them drops to one-fourth of its initial value.

2. Value of the Proportionality Constant ($k$): In a vacuum or air, the constant $k = \frac{1}{4\pi\varepsilon_0}$ has an experimentally determined value of approximately $9 \times 10^9\text{ N m}^2\text{ C}^{-2}$.

3. Definition of $1\text{ Coulomb}$: Using Coulomb's law, if $q_1 = q_2 = 1\text{ C}$ and $r = 1\text{ m}$ in a vacuum, the force becomes:

Therefore, $1\text{ Coulomb}$ is defined as the amount of charge that repels an identical charge placed $1\text{ meter}$ away in a vacuum with a force of $9 \times 10^9\text{ Newtons}$.

4. Influence of the Medium: If the charges are immersed in a material medium (such as water or glass) with permittivity $\varepsilon$, the force becomes:

Since $\varepsilon = \varepsilon_r \varepsilon_0$ (where $\varepsilon_r > 1$ is the relative permittivity), the electrostatic force decreases in a material medium compared to a vacuum:

5. Newton's Third Law Verification: The force exerted on $q_1$ by $q_2$ ($\vec{F}_{12}$) uses a unit vector $\hat{r}_{21}$ which points in the opposite direction to $\hat{r}_{12}$. Therefore:

This demonstrates that the electrostatic forces form an action-reaction pair, complying with Newton’s third law.

#### Definition

The electric field ($\vec{E}$) at a given point in space is defined as the electrostatic force experienced per unit positive test charge ($q_0$) placed at that point.

If the electric field is produced by a single source point charge $q$, the force on a test charge $q_0$ at a distance $r$ is given by Coulomb's law. Substituting that force into the field definition yields:

Where $\hat{r}}$ is the unit vector pointing from the source charge $q$ toward the position of the test charge. The SI unit of electric field is Newton per Coulomb ($\text{N C}^{-1}$) or Volt per meter ($\text{V m}^{-1}$).

#### Various Aspects of the Electric Field

1. Independent of the Test Charge: Even though the electric field is defined using a test charge $q_0$, the field value is mathematically independent of $q_0$. It depends exclusively on the magnitude of the source charge $q$ and its distance $r$.

2. Direction Convention: * If the source charge $q$ is positive, the electric field vectors point radially outwards away from the charge.

• If the source charge $q$ is negative, the electric field vectors point radially inwards toward the charge.

3. Force Calculation: If an electric field $\vec{E}$ is known in a region, the electrostatic force $\vec{F}$ acting on any arbitrary charge $Q$ brought into that region can be directly determined using:

4. Superposition Principle Compatibility: The electric field satisfies the principle of superposition. For a collection of discrete source charges $q_1, q_2, \dots, q_n$, the total electric field at a point is the vector sum of the independent electric fields produced by each charge:

5. Physical Reality and Finite Velocity: The electric field is not a mere mathematical convenience; it is a real physical entity that possesses energy and momentum. When a charge is accelerated, it produces electromagnetic disturbances that propagate through the electric field at the speed of light ($c$), confirming that electrostatic interactions are not instantaneous "action-at-a-distance" phenomena.

An electric dipole consists of two equal and opposite charges, $-q$ and $+q$, separated by a small distance $2a$. Let the dipole moment be $\vec{p} = 2qa\,\hat{p}$, where $\hat{p}$ is a unit vector pointing from $-q$ to $+q$.

#### Part A: Electric Field on the Axial Line

Consider a point $C$ lying on the dipole axis at a distance $r$ from its midpoint $O$, on the side of $+q$.

1. Electric field at $C$ due to $+q$ ($\vec{E}_+$):

The distance from $+q$ to $C$ is $(r - a)$. Since $+q$ is a positive charge, the field points away from it (along $\hat{p}$):

2. Electric field at $C$ due to $-q$ ($\vec{E}_-$):

The distance from $-q$ to $C$ is $(r + a)$. Since $-q$ is a negative charge, the field points toward it (opposite to $\hat{p}$):

3. Total Electric Field ($\vec{E}_{\text{total}}$):

By the superposition principle, the total field at $C$ is:

Simplifying the bracketed term using a common denominator:

4. Short Dipole Approximation ($r \gg a$):

If the point $C$ is very far away, we can neglect $a^2$ compared to $r^2$ ($r^2 - a^2 \approx r^2$). Thus, $(r^2 - a^2)^2 \approx r^4$:

Substituting $p = 2qa$:

The total electric field on the axial line points in the direction of the dipole moment vector $\vec{p}$.

---

#### Part B: Electric Field on the Equatorial Plane

Consider a point $C$ lying on the equatorial plane (perpendicular bisector) at a distance $r$ from the midpoint $O$.

1. Magnitudes of individual fields:

The distance from both $+q$ and $-q$ to the point $C$ is identical, given by the Pythagorean theorem as $d = \sqrt{r^2 + a^2}$. Therefore, the magnitudes of the electric fields due to individual charges are equal:

2. Resolution into Components:

Let $\theta$ be the angle made by the lines joining the charges to point $C$ with the dipole axis.

• The components perpendicular to the dipole axis ($E_+\sin\theta$ and $E_-\sin\theta$) are equal in magnitude and opposite in direction, so they cancel out.
• The components parallel to the dipole axis ($E_+\cos\theta$ and $E_-\cos\theta$) point in the same direction, which is opposite to the dipole moment vector $\hat{p}$.

3. Total Electric Field ($\vec{E}_{\text{total}}$):

From the geometry of the triangle, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{\sqrt{r^2 + a^2}}$. Substituting $E_+$ and $\cos\theta$:

4. Short Dipole Approximation ($r \gg a$):

Neglecting $a^2$ relative to $r^2$, the term $(r^2 + a^2)^{3/2}$ simplifies to $(r^2)^{3/2} = r^3$:

The electric field on the equatorial plane is opposite in direction to the electric dipole moment vector $\vec{p}$, and its magnitude is exactly half of the field at an equidistant axial point.

Consider an electric dipole consisting of two equal and opposite charges $-q$ and $+q$ separated by a distance $2a$. Let this dipole be placed in a uniform external electric field $\vec{E}$ such that the dipole axis makes an angle $\theta$ with the direction of the electric field.

#### 1. Forces acting on the charges

• The positive charge $+q$ experiences an electrostatic force $\vec{F}_+ = +q\vec{E}$ acting along the direction of the electric field.
• The negative charge $-q$ experiences an electrostatic force $\vec{F}_- = -q\vec{E}$ acting in the direction opposite to the electric field.

#### 2. Net Translational Force

The net translational force acting on the dipole is the vector sum of these individual forces:

Because the net force is zero, the dipole will not undergo any linear or translational acceleration in a uniform electric field.

#### 3. Calculation of Torque

Even though the net force is zero, the two forces $+q\vec{E}$ and $-q\vec{E}$ have different lines of action. This configuration forms a couple, which exerts a net torque on the dipole, causing it to rotate.

The magnitude of the torque ($\tau$) is equal to the magnitude of one of the forces multiplied by the perpendicular distance between their lines of action:

From the geometry of the system, if $2a$ is the length of the dipole, the perpendicular distance between the lines of action of the two forces is given by $2a\sin\theta$.

Since the magnitude of the electric dipole moment is $p = 2qa$, substituting this gives:

#### 4. Vector Form

The direction of the torque is perpendicular to both the dipole moment $\vec{p}$ and the electric field $\vec{E}$, acting to align the dipole with the field lines. It can be expressed via the vector cross-product:

#### 5. Special Cases

• Case 1 (Minimum Torque): When $\theta = 0^\circ$ or $180^\circ$, $\sin\theta = 0$, so $\tau = 0$. The dipole is in stable ($\theta=0^\circ$) or unstable ($\theta=180^\circ$) equilibrium.
• Case 2 (Maximum Torque): When $\theta = 90^\circ$, $\sin(90^\circ) = 1$, so $\tau_{\text{max}} = pE$. The torque is at its maximum value when the dipole is aligned perpendicular to the field.

Consider a stationary positive point charge $+q$ placed at the origin $O$. We wish to determine the electrostatic potential $V$ at a point $P$ located at a distance $r$ from the origin.

#### 1. Definition and Setting up the Integral

By definition, the electric potential $V$ at point $P$ is equal to the work done by an external agent in moving a unit positive test charge ($+1\text{ C}$) from infinity ($\infty$) to the point $P$ at constant velocity:

Where $\vec{E}$ is the electric field produced by the point charge $+q$, and $d\vec{r}$ is an infinitesimal displacement along the radial path.

#### 2. Substituting the Electric Field

The electric field $\vec{E}$ at an intermediate point along the path at a distance $r'$ from the origin is given by Coulomb's law:

Since the displacement $d\vec{r} = dr'\,\hat{r}$ is along the radial line, the dot product $\vec{E} \cdot d\vec{r}$ simplifies to:

#### 3. Evaluation of the Integral

Substitute this expression back into the potential formula and set the integration limits from infinity ($\infty$) to the final distance ($r$):

Take the constant terms out of the integral:

Using the integration power rule ($\int x^{-2}dx = -x^{-1} = -\frac{1}{x}$):

Applying the upper and lower boundary limits:

Since $\frac{1}{\infty} = 0$, the expression reduces to:

#### Important Conclusions

• If the source charge is positive ($+q$), the potential is positive ($V > 0$).
• If the source charge is negative ($-q$), the potential is negative ($V = -\frac{1}{4\pi\varepsilon_0} \frac{q}{r}$).
• The potential decreases inversely with distance ($V \propto \frac{1}{r}$), which is slower than the inverse-square decrease of the electric field ($E \propto \frac{1}{r^2}$).

Consider an electric dipole consisting of two equal and opposite point charges, $-q$ and $+q$, separated by a small distance $2a$. Let $O$ be the midpoint of the dipole. We wish to calculate the electrostatic potential $V$ at an arbitrary point $P$ located at a distance $r$ from $O$, where the line $OP$ makes an angle $\theta$ with the dipole axis (on the side of $+q$).

Let $r_1$ be the distance from $+q$ to $P$, and $r_2$ be the distance from $-q$ to $P$.

1. Total Potential by Superposition Principle:

The total electric potential at $P$ is the scalar sum of the potentials due to individual charges $+q$ and $-q$:

2. Finding $\frac{1}{r_1}$ and $\frac{1}{r_2}$ using the Law of Cosines:

From the geometry of the triangle formed by charges and point $P$:

• For triangle $OP(+q)$, using the cosine rule:

Factoring out $r^2$:

Since $r \gg a$, the term $\frac{a^2}{r^2}$ is extremely small and can be neglected:

Taking the square root on both sides:

Taking the reciprocal and applying the binomial theorem ($(1 - x)^{-1/2} \approx 1 + \frac{1}{2}x$):

• For triangle $OP(-q)$, the angle is $(180^\circ - \theta)$, and $\cos(180^\circ - \theta) = -\cos\theta$:

Neglecting $\frac{a^2}{r^2}$ and applying the binomial theorem in a similar manner:

3. Substituting back into the Potential Expression:

4. In Terms of Dipole Moment:

Since the electric dipole moment magnitude is $p = 2qa$:

In vector notation, using the unit vector $\hat{r}$ pointing along $OP$:

5. Special Cases:

• Case 1: If point $P$ lies on the axial line on the side of $+q$ ($\theta = 0^\circ \implies \cos 0^\circ = 1$):

• Case 2: If point $P$ lies on the axial line on the side of $-q$ ($\theta = 180^\circ \implies \cos 180^\circ = -1$):

• Case 3: If point $P$ lies on the equatorial plane ($\theta = 90^\circ \implies \cos 90^\circ = 0$):

The electrostatic potential energy of a system of charges is equal to the total work done by an external agent in assembling the charges together by bringing them from infinity to their respective positions without acceleration.

Consider three point charges $q_1$, $q_2$, and $q_3$ to be brought from infinity to positions $A$, $B$, and $C$ respectively. Let the distance between $q_1$ and $q_2$ be $r_{12}$, between $q_2$ and $q_3$ be $r_{23}$, and between $q_1$ and $q_3$ be $r_{13}$.

1. Bringing the first charge $q_1$:

Since there is no existing electric field in space initially, no electrostatic force opposes the movement.

2. Bringing the second charge $q_2$:

When $q_2$ is brought from infinity to position $B$ at a distance $r_{12}$ from $A$, it moves against the electric field produced by $q_1$. The potential at $B$ due to $q_1$ is:

3. Bringing the third charge $q_3$:

When $q_3$ is brought from infinity to position $C$, it moves against the combined electric fields produced by both $q_1$ and $q_2$. The total potential at $C$ due to $q_1$ and $q_2$ is:

4. Total Potential Energy ($U$):

The total electrostatic potential energy of the system is the sum of the work done in each individual step:

Consider an electric dipole with dipole moment $\vec{p}$ placed in a uniform external electric field $\vec{E}$ at an angle $\theta$. The external field exerts a torque $\tau = pE\sin\theta$ that rotates the dipole to align it with the field direction.

To find the electrostatic potential energy stored in this alignment, we calculate the work done by an external agent to rotate the dipole against this torque.

1. Infinitesimal Work Done ($dW$):

The small amount of work done in rotating the dipole through an infinitesimal angle $d\theta$ against the restoring electrostatic torque is:

2. Total Work Done ($W$):

The total work done in rotating the dipole from an initial configuration angle $\theta'$ to a final angle $\theta$ is found by integrating the expression:

3. Potential Energy Definition ($U$):

By physical convention, the potential energy is chosen to be zero when the dipole is perpendicular to the field ($\theta' = 90^\circ \implies \cos 90^\circ = 0$). This work done is stored as potential energy $U(\theta)$:

4. Vector Form:

Using the vector dot product definition, this scalar expression can be written cleanly as:

5. Analysis of Orientations:

• Stable Equilibrium ($\theta = 0^\circ$): $U = -pE\cos(0^\circ) = -pE$. The potential energy is minimum when the dipole aligns exactly parallel to the field lines.
• Unstable Equilibrium ($\theta = 180^\circ$): $U = -pE\cos(180^\circ) = +pE$. The potential energy is at its maximum absolute value when the dipole is aligned antiparallel.

Consider a single isolated positive point charge $+q$ located at the origin $O$. To determine the total electric flux passing through a closed space, imagine a spherical surface of radius $r$ centered at the origin enclosing this charge. This hypothetical surface acts as our Gaussian surface.

1. Electric Field at the Surface:

According to Coulomb's law, the electric field $\vec{E}$ at any point on this spherical surface at a distance $r$ from the charge is:

Where $\hat{r}$ is a radial unit vector pointing outwards from the center.

2. Area Element Vector:

Consider an infinitesimal area element $dA$ on the spherical surface. The area vector $d\vec{A}$ points normally outward from the surface, which means it is parallel to the radial direction:

3. Flux through the Area Element ($d\Phi_E$):

The small electric flux passing through this area element is given by the dot product:

Since $\hat{r} \cdot \hat{r} = 1$:

4. Total Electric Flux ($\Phi_E$):

The total electric flux through the entire closed spherical surface is found by integrating $d\Phi_E$ over the surface:

Since the terms $\frac{1}{4\pi\varepsilon_0}$ and the distance $r$ are constant everywhere on the sphere, they can be pulled out of the closed surface integral:

The term $\oint_S dA$ represents the total surface area of a sphere of radius $r$, which is equal to $4\pi r^2$:

Canceling out the common terms $4\pi$ and $r^2$ from the numerator and denominator simplifies the expression to:

This result confirms Gauss's law, proving that the total electric flux through any closed surface is equal to $\frac{1}{\varepsilon_0}$ times the net charge enclosed within that surface.

Consider an infinitely long, straight wire carrying a uniform linear charge density $\lambda$ (charge per unit length). Due to the cylindrical symmetry of the wire, the electric field $\vec{E}$ must point radially outwards perpendicular to the wire at every point, and its magnitude depends only on the radial distance $r$.

1. Choosing the Gaussian Surface:

We choose a closed cylindrical Gaussian surface of radius $r$ and length $L$ coaxially surrounding the wire. This cylinder has three distinct surfaces:

• Two flat circular end caps (Top and Bottom surfaces).
• One curved lateral surface.

2. Evaluating the Flux Integral ($\oint \vec{E} \cdot d\vec{A}$):

The total electric flux through this closed cylinder is broken down into three parts:

• For the Top and Bottom flat surfaces: The electric field $\vec{E}$ is directed radially outward, while the area vector $d\vec{A}$ is perpendicular to the end caps (pointing vertically out). The angle between $\vec{E}$ and $d\vec{A}$ is $90^\circ$. Since $\cos 90^\circ = 0$:

• For the Curved surface: The electric field $\vec{E}$ and the area vector $d\vec{A}$ both point radially outwards in the same direction, so the angle between them is $0^\circ$. Since $\cos 0^\circ = 1$:

The total surface area of the curved portion of a cylinder is $2\pi rL$. Therefore:

3. Determining the Enclosed Charge ($Q_{\text{encl}}$):

The length of the wire contained within the Gaussian cylinder is $L$. Since the linear charge density is $\lambda$:

4. Applying Gauss's Law:

Canceling the length $L$ from both sides and solving for $E$:

5. Vector Form:

Where $\hat{r}$ is the unit vector pointing radially outward from the wire.

Consider an infinite, thin plane sheet of charge with a uniform surface charge density $\sigma$ (charge per unit area). Due to planar symmetry, the electric field $\vec{E}$ must be uniform in magnitude, point normally away from the sheet on both sides (if $\sigma$ is positive), and be independent of the distance from the sheet.

1. Choosing the Gaussian Surface:

We choose a cylindrical Gaussian surface (or rectangular box) of cross-sectional area $A$ and total length $2r$, positioned such that it passes through the sheet perpendicularly, with its flat end faces parallel to the sheet at an equal distance $r$ on either side.

2. Evaluating the Flux Integral ($\oint \vec{E} \cdot d\vec{A}$):

The total flux through this closed surface consists of contributions from the curved walls and the two flat end caps:

• For the Curved wall: The electric field lines run parallel to the curved surface, meaning the area vector $d\vec{A}$ is perpendicular to $\vec{E}$ ($\theta = 90^\circ$). Thus, the flux through the curved wall is zero:

• For the Flat caps: On both the left and right end caps, the electric field lines pass straight out through the faces, running parallel to the outward area vector $d\vec{A}$ ($\theta = 0^\circ$). Since the electric field magnitude $E$ is identical at both caps:

Summing these parts gives the total flux:

3. Determining the Enclosed Charge ($Q_{\text{encl}}$):

The Gaussian cylinder intercepts an area $A$ on the charged sheet. Given the surface charge density $\sigma$:

4. Applying Gauss's Law:

The area term $A$ cancels out from both sides, leaving:

5. Vector Form:

Where $\hat{n}$ is a unit vector pointing normally away from the plane sheet on either side.

Consider a thin, hollow spherical shell of radius $R$ carrying a total charge $Q$ uniformly distributed over its surface. The surface charge density is $\sigma = \frac{Q}{4\pi R^2}$. To find the electric field $\vec{E}$ at any point $P$ at a distance $r$ from the center $O$, we utilize a spherical Gaussian surface of radius $r$ concentric with the shell.

Due to spherical symmetry, the electric field $\vec{E}$ points radially outwards (if $Q > 0$) or radially inwards (if $Q < 0$), and its magnitude is constant at all points on the chosen Gaussian surface. The area vector $d\vec{A}$ of any small element on this sphere also points radially outward, meaning $\vec{E}$ and $d\vec{A}$ are parallel ($\theta = 0^\circ$).

The total electric flux $\Phi_E$ through the Gaussian surface is:

Since the total surface area of the spherical Gaussian surface is $4\pi r^2$:

#### Case 1: At a point outside the shell ($r > R$)

The Gaussian surface encloses the entire charge $Q$ distributed on the shell ($Q_{\text{encl}} = Q$).

Applying Gauss's law:

In vector form:

Thus, for points outside the shell, the electric field behaves exactly as if the entire charge $Q$ were concentrated as a point charge at the center $O$.

#### Case 2: At a point on the surface of the shell ($r = R$)

Substituting $r = R$ into the outside field formula gives:

This is the maximum value of the electric field produced by the shell.

#### Case 3: At a point inside the shell ($r < R$)

When the point $P$ lies inside the shell, the spherical Gaussian surface of radius $r$ is entirely inside the hollow cavity. Because all the charges reside exclusively on the outer surface of the shell, the net charge enclosed by this inner Gaussian surface is exactly zero ($Q_{\text{encl}} = 0$).

Applying Gauss's law:

Thus, the electric field at all points inside a uniformly charged hollow spherical shell is zero.

A conductor contains a large number of mobile charge carriers (free electrons). In electrostatic equilibrium, these charges have completely finished rearranging themselves, and there is no net motion of charge. Under these conditions, conductors exhibit the following essential properties:

#### 1. The net electric field inside the conductor is zero

If an external electric field $\vec{E}_0$ is applied to a conductor, the free electrons quickly drift in the opposite direction of the field. They accumulate on one side, leaving a positive charge accumulation on the opposite side. This process sets up an internal induced electric field $\vec{E}_{\text{ind}}$ that opposes the external field. The migration continues until the magnitude of the induced field perfectly matches the external field:

#### 2. The net charge density inside the conductor is zero; any excess charge resides exclusively on the surface

According to Gauss's law, the total electric flux through any arbitrary surface drawn inside the conductor is proportional to the enclosed charge. Since the electric field $\vec{E}$ is zero everywhere inside the conductor:

Consequently, any excess static charge placed on a conductor is repelled to its outermost boundary.

#### 3. The electric field just outside the surface of a conductor is perpendicular to the surface at every point

If the electric field vector $\vec{E}$ had a tangential component parallel to the surface, it would exert a force ($F = qE_{\text{tangential}}$) on the surface charges, causing them to move along the surface. This would violate the condition of electrostatic equilibrium. Therefore, the tangential component must be zero, forcing the electric field to be entirely normal to the surface:

Where $\sigma$ is the local surface charge density.

#### 4. The electrostatic potential is constant throughout the volume of the conductor and matches its value on the surface

The relationship between the electric field and potential difference between any two internal points $A$ and $B$ is given by:

Since $\vec{E} = 0$ everywhere inside the conductor, the integral vanishes ($V_B - V_A = 0$), meaning $V_A = V_B$. Thus, the entire volume of a static conductor forms an equipotential region.

#### 5. The electric field inside a hollow cavity of a conductor is zero

Regardless of the presence of external fields or charges on the conductor's outer boundary, if a hollow cavity contains no net charge inside it, the electric field within that cavity is zero. This principle forms the foundation for electrostatic shielding.

Electrostatic induction is the process of charging an uncharged conducting body by bringing a charged object close to it, without making any physical contact between the two bodies.

The complete step-by-step mechanism can be understood using an isolated neutral metallic sphere mounted on an insulating stand:

1. Step 1: Bringing a charged body close

Initially, the metal sphere is electrically neutral, meaning it contains an equal number of positive and negative charges uniformly distributed. When a negatively charged plastic rod is brought close to (but not touching) the sphere, the free electrons in the metal sphere experience a repulsive force. They migrate to the far side of the sphere, creating a net accumulation of negative charge on the distant edge and leaving a net accumulation of positive charge on the near edge.

2. Step 2: Grounding the conductor

While keeping the negatively charged rod fixed in its position, the far side of the sphere is connected to the earth using a conducting wire (grounding). The accumulated free electrons on the far side flow through the wire into the ground. The positive charges on the near side remain bound in place by the strong attractive electrostatic force exerted by the negative rod.

3. Step 3: Disconnecting the ground

The grounding wire is disconnected from the sphere while keeping the charged rod in place. The positive charges remain localized on the side near the rod.

4. Step 4: Removing the charged body

Finally, the negatively charged rod is moved far away from the sphere. With the external field removed, the bound positive charges are no longer held on one side. Due to mutual electrostatic repulsion, they redistribute themselves uniformly across the entire surface of the metallic sphere.

As a result, the metallic sphere acquires a permanent net positive charge without ever having touched the charging rod. If a positively charged rod were used initially, the process would instead leave the sphere with a net negative charge.

A dielectric is a non-conducting material (insulator) that does not possess any free electrons to conduct electricity. However, when placed inside an external electric field, the charges within its molecules redistribute, leading to the creation of an induced internal electric field. Dielectrics are broadly classified into two main categories:

#### 1. Non-polar Dielectric Molecules

In non-polar molecules, the centers of positive charge (atomic nuclei) and negative charge (electron cloud) coincide with each other under normal circumstances. Because there is no separation between the charges, these molecules have zero inherent permanent electric dipole moment. Examples include Hydrogen ($H_2$), Nitrogen ($N_2$), and Oxygen ($O_2$).

#### 2. Polar Dielectric Molecules

In polar molecules, due to asymmetric arrangements or differences in electronegativity, the centers of positive and negative charges are permanently separated by a small distance even in the absence of an external electric field. Consequently, they possess an inherent permanent electric dipole moment. However, in bulk material, these individual dipoles are randomly oriented due to thermal agitation, resulting in a net macroscopic dipole moment of zero. Examples include Water ($H_2O$), Ammonia ($NH_3$), and Hydrogen Chloride ($HCl$).

#### Mechanism of Induced Electric Field inside a Dielectric

When any dielectric slice is placed into a uniform external electric field ($\vec{E}_0$):

1. Alignment of Dipoles: The external field exerts a torque on the atomic or molecular structures.

• In non-polar dielectrics, the external field shifts the positive and negative charge centers in opposite directions, inducing a tiny dipole moment in each molecule.
• In polar dielectrics, the external field aligns the pre-existing randomly oriented permanent molecular dipoles along the direction of the field lines.

2. Induced Surface Charges: Inside the bulk of the dielectric material, the positive end of one molecular dipole sits right next to the negative end of its neighbor, effectively canceling each other out. However, at the exterior boundaries of the dielectric, this cancellation does not occur:

• A net layer of negative induced bound charge accumulates on the surface facing the positive electrode.
• A net layer of positive induced bound charge accumulates on the surface facing the negative electrode.

3. Creation of the Internal Field: These induced surface charges set up an internal induced electric field ($\vec{E}_{\text{ind}}$) directed from the induced positive surface to the induced negative surface. This induced field points in the opposite direction of the external field ($\vec{E}_0$).

4. Net Internal Field: The total resultant electric field ($\vec{E}_{\text{net}}$) inside the dielectric material is the vector sum of both fields:

Because these charges are bound to their molecules and cannot move completely freely like electrons in a metal, the induced field $\vec{E}_{\text{ind}}$ is always weaker than the external field $\vec{E}_0$. Thus, the presence of a dielectric weakens, but does not completely eliminate, the electric field inside its volume.

A parallel plate capacitor consists of two identical conducting metal plates, each of area $A$, separated by a small distance $d$. Let the space between the plates be filled with air or vacuum.

1. Electric Field between the Plates:

Suppose the plates are given equal and opposite charges, $+Q$ and $-Q$. This creates a uniform surface charge density $\sigma = \frac{Q}{A}$ on the positive plate and $-\sigma = -\frac{Q}{A}$ on the negative plate.

• In the regions outside the top and bottom plates, the fields due to the two individual plates point in opposite directions and cancel out ($E = 0$).
• In the region between the plates, the electric field produced by the positive plate and the negative plate point in the same direction (from the positive plate toward the negative plate). Summing them together gives:

Substituting $\sigma = \frac{Q}{A}$:

2. Potential Difference between the Plates:

The potential difference $V$ between two plates separated by a uniform electric field $E$ over a distance $d$ is given by:

Substituting the expression for the electric field $E$ into this equation:

3. Capacitance Calculation:

By definition, the capacitance $C$ of a capacitor is the ratio of the charge stored $Q$ to the potential difference $V$:

Substituting the calculated value of $V$:

The charge term $Q$ cancels out from both the numerator and denominator, leaving:

This mathematical expression shows that the capacitance of a parallel plate capacitor is directly proportional to the surface area $A$ of its plates and inversely proportional to the separation distance $d$ between them.

A capacitor is a device used to store electrical charge and electrical potential energy. The process of charging a capacitor involves transferring electrons from one plate to the other, which requires doing work against the growing electrostatic repulsive forces. This work done is stored as electrostatic potential energy within the electric field between the plates.

Consider a capacitor of capacitance $C$. Let it be partially charged to an intermediate charge state $q$, creating an intermediate potential difference $v = \frac{q}{C}$ between its plates.

1. Infinitesimal Work Done ($dW$):

The small amount of work done $dW$ by an external source to transfer an additional infinitesimal charge element $dq$ from the negative plate to the positive plate against this potential difference is:

2. Total Work Done ($W$):

The total work done to fully charge the capacitor from an initially uncharged state ($q = 0$) to a final total charge state ($q = Q$) is obtained by integrating this expression:

Taking the constant capacitance $C$ outside the integral:

Using the integration power rule ($\int q \, dq = \frac{q^2}{2}$):

3. Alternative Expressions for Energy ($U$):

Since this work done is perfectly stored as electrostatic potential energy ($U = W$), we have:

Using the fundamental capacitor relationship $Q = CV$, we can substitute for $Q$ or $C$ to obtain two alternative formats:

• Substituting $Q = CV$:

• Substituting $C = \frac{Q}{V}$:

The effect of introducing a dielectric material between the plates of a parallel plate capacitor depends on whether the capacitor remains connected to the charging battery or is disconnected before the dielectric is inserted.

#### Case 1: When the capacitor is disconnected from the battery

Consider a capacitor charged by a battery to a charge $Q_0$, potential difference $V_0$, and electric field $E_0$. The energy stored is $U_0 = \frac{1}{2}Q_0V_0$. The battery is then disconnected, and a dielectric slab of relative permittivity $\varepsilon_r$ is inserted to completely fill the space.

1. Charge ($Q$): Since the battery is disconnected, the plates are electrically isolated. According to the law of conservation of charge, the charge remains constant:

2. Electric Field ($E$): The electric field polarizes the dielectric, creating an opposing internal field. The net electric field decreases by a factor of $\varepsilon_r$:

3. Potential Difference ($V$): Since $V = Ed$, a decrease in the electric field leads to a proportional drop in voltage:

4. Capacitance ($C$): Using the relationship $C = \frac{Q}{V}$:

The capacitance increases by a factor of $\varepsilon_r$.

5. Energy Storing ($U$): The new energy stored is:

The potential energy decreases because work is done by the electrostatic field to pull the dielectric slab into the plates.

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#### Case 2: When the capacitor remains connected to the battery

Consider the same capacitor, but in this case, the charging battery remains connected while the dielectric slab is inserted.

1. Potential Difference ($V$): Because the capacitor remains connected to the battery, the voltage across the plates is held fixed at its initial value:

2. Capacitance ($C$): The capacitance depends on the geometry and material properties, so it increases by a factor of $\varepsilon_r$ exactly as before:

3. Charge ($Q$): Since $Q = CV$ and $C$ has increased while $V$ is constant, the battery supplies additional charges to the plates:

The charge increases by a factor of $\varepsilon_r$.

4. Electric Field ($E$): Since $E = \frac{V}{d}$ and both $V$ and $d$ are unchanged, the electric field remains the same:

5. Energy Storing ($U$): The new energy stored is:

The stored energy increases because the battery performs work to transfer extra charges onto the plates.

#### Part A: Capacitors in Series

Consider three capacitors of capacitances $C_1$, $C_2$, and $C_3$ connected end-to-end in a single line across a battery providing a potential difference $V$.

1. Distribution of Charge and Voltage:

• In a series circuit, the charge $Q$ flowing through each capacitor is identical because there is only one path for the current to take: $Q_1 = Q_2 = Q_3 = Q$.
• The total potential difference $V$ supplied by the battery is divided across the individual capacitors:

2. Applying the Capacitance Formula ($V = \frac{Q}{C}$):

The individual voltages across each capacitor are:

If $C_s$ represents the equivalent (resultant) capacitance of the entire series combination, then:

3. Derivation:

Substitute these expressions into the total voltage equation:

Factoring out and canceling the common charge term $Q$ from both sides gives:

Thus, the reciprocal of the equivalent capacitance of capacitors connected in series is equal to the sum of the reciprocals of their individual capacitances. The value of $C_s$ is always smaller than the smallest individual capacitance in the combination.

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#### Part B: Capacitors in Parallel

Consider three capacitors of capacitances $C_1$, $C_2$, and $C_3$ connected side-by-side with their corresponding plates joined to common junctions across a battery providing a potential difference $V$.

1. Distribution of Charge and Voltage:

• Since all capacitors are connected directly across the same terminals of the battery, the potential difference across each of them is the same: $V_1 = V_2 = V_3 = V$.
• The total charge $Q$ pulled from the battery splits among the parallel branches:

2. Applying the Capacitance Formula ($Q = CV$):

The individual charges stored on each capacitor are:

If $C_p$ represents the equivalent (resultant) capacitance of the entire parallel combination, then:

3. Derivation:

Substitute these expressions into the total charge equation:

Factoring out and canceling the common voltage term $V$ from both sides gives:

Thus, the equivalent capacitance of capacitors connected in parallel is equal to the direct sum of their individual capacitances. The value of $C_p$ is always larger than the largest individual capacitance in the combination.

#### 1. Distribution of Charges in an Arbitrarily Shaped Conductor

For a perfectly spherical conductor, excess electrostatic charges distribute themselves completely uniformly over the surface because of symmetry. However, for an irregularly shaped conductor, the surface charge density $\sigma$ (charge per unit area) varies across different parts of the geometry.

[Image showing charge distribution on an irregularly shaped conductor, with a high concentration of charges at sharp points]

Let two conducting spheres of different radii $r_1$ and $r_2$ be connected to each other by a long thin conducting wire. Since they are connected, charges flow between them until both spheres reach the exact same electrostatic potential ($V_1 = V_2 = V$).

The surface charge density $\sigma$ is defined as $\sigma = \frac{q}{4\pi r^2}$, which means $q = \sigma(4\pi r^2)$. Substituting this into the ratio yields:

This relationship shows that the surface charge density is inversely proportional to the radius of curvature. At sharp points or jagged edges, the radius of curvature $r$ approaches a very small value, causing the local charge density $\sigma$ to become exceptionally high.

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#### 2. Principle behind the Lightning Conductor

A lightning conductor (or lightning rod) is a safety device used to protect tall buildings from being damaged by lightning strikes. It operates based on two primary principles: action of points (corona discharge) and electrostatic induction.

• Construction: It consists of a long, thick copper rod running down the side of a building. The top end of the rod features several sharp, pointed spikes pointing toward the sky, while the bottom end is connected to a heavy copper plate buried deep inside the earth.
• Working Mechanism:

1. When a highly charged storm cloud (e.g., carrying a net negative charge) passes directly over the building, it induces an opposite positive charge on the sharp spikes of the lightning conductor through electrostatic induction.

2. Due to the action of points, the extremely small radius of curvature at the sharp tips creates an intense localized electric field. This field ionizes the surrounding air molecules into positive and negative ions.

3. The negative ions are repelled by the cloud and attracted to the spikes, where they discharge safely into the earth. Simultaneously, the positive ions are repelled by the spikes toward the cloud, neutralizing a portion of the cloud's negative charge. This process reduces the likelihood of a disruptive lightning strike.

4. If a lightning strike does occur, the highly conductive copper rod provides a path of least resistance, safely guiding the massive electrical current down into the earth and protecting the structural integrity of the building.

A Van de Graaff generator is an electrostatic machine designed by Robert J. Van de Graaff in 1929. It is capable of producing exceptionally high electrostatic potential differences (on the order of several million volts, $10^7\text{ V}$).

#### 1. Principle of Operation

The design and operation of a Van de Graaff generator are based on two fundamental electrostatic phenomena:

• Corona Discharge (Action of Points): Electric charge leaks out or ionizes the surrounding air rapidly from the sharp pointed ends of a highly charged conductor.
• Electrostatic Shielding/Property of Conductors: When an internal charged conductor touches the inner wall of a hollow spherical conductor, its entire excess charge is transferred completely to the outer surface of the hollow sphere, regardless of how much charge is already present there.

#### 2. Construction

• Hollow Metallic Sphere ($A$): A large, hollow spherical conductor is mounted on top of tall, insulating pillars.
• Pulleys ($B$ and $C$): A pulley $B$ is fixed at the center of the hollow sphere, and another pulley $C$ is fixed at the bottom, near the ground. A continuous conveyor belt made of an insulating material (such as rubber or silk) loops around both pulleys. The lower pulley $C$ is continuously rotated by an electric motor.
• Metallic Combs ($D$ and $E$): Two comb-shaped metallic conductors featuring numerous sharp teeth are positioned near the belt:
• Spray Comb ($D$): Located near the lower pulley $C$, outside the sphere. It is connected to a high-voltage rectified power supply ($10^4\text{ V}$).
• Collecting Comb ($E$): Located near the upper pulley $B$, inside the hollow metallic sphere. It is directly connected to the inner surface of the sphere.

#### 3. Working Mechanism

1. Spraying of Charge: The high-voltage power supply maintains the spray comb $D$ at a very high positive potential. Due to the action of points (corona discharge), the intense electric field near the sharp teeth of comb $D$ ionizes the surrounding air. The positive ions are repelled by the comb and stick to the moving insulating belt.

2. Transportation: The electric motor turns the lower pulley $C$, moving the belt upward. The belt carries these positive charges from the bottom upward into the hollow sphere.

3. Collection of Charge: As the charged portion of the belt reaches the top and passes close to the collecting comb $E$, the positive charges on the belt induce a negative charge on the sharp teeth of comb $E$ and an equal positive charge on the outer surface of the hollow sphere $A$ through electrostatic induction.

4. Discharge at the Upper Comb: The high localized field at the teeth of comb $E$ causes a corona discharge that ionizes the air inside the sphere. The negative ions are repelled toward the belt, completely neutralizing its positive charges before it loops down. The uncharged belt moves downward to receive a fresh layer of charge at comb $D$.

5. Accumulation of High Voltage: This process continues systematically. Since any charge transferred to the collecting comb immediately flows to the outer surface of the sphere, the charge on the sphere builds up continuously. As a result, the electrostatic potential ($V = \frac{Q}{4\pi\varepsilon_0 R}$) of the sphere rises until it reaches the breakdown limit of the surrounding air.

#### 4. Preventing Leakage

When the electric field around the outer surface of the sphere exceeds the dielectric strength of air ($\approx 3 \times 10^6\text{ V m}^{-1}$), the air undergoes breakdown, and charges begin to leak into the surroundings. To minimize this leakage and allow the potential to build up to several million volts, the entire generator assembly is enclosed within a steel tank filled with a gas (such as air or nitrogen) at very high pressure.

#### 5. Applications

• The massive potential differences generated are used to accelerate charged particles (such as protons, deuterons, and alpha particles) to very high kinetic energies.
• These highly energetic beams of particles are utilized to trigger nuclear disintegrations and study the fundamental structure of atomic nuclei in nuclear physics research.