For a semicircular current of radius r, the magnetic field at the centre is $B=\dfrac{\mu_0 I}{4r}$. (This follows from Biot–Savart law for an arc of angle $\pi$.)
Undeflected motion requires magnetic force = electric force: $evB=eE$. Thus $v=E/B$. For infinite charged plates $E=\sigma/\varepsilon_0$, so $v=\sigma/(\varepsilon_0 B)$. Time to cross distance $l$ is $t=l/v=\varepsilon_0 l B/\sigma$.
Kinetic energy after acceleration: $\tfrac12 mv^2=qV\Rightarrow v=\sqrt{2qV/m}$. Magnetic force (perpendicular case) $F=qvB=qB\sqrt{2qV/m}$.
Magnetic dipole moment $\mu=N I A$. Here $N=50$, $I=3\,$A, $A=\pi r^2=\pi(0.05)^2\approx7.854\times10^{-3}\,$m^2. So $\mu\approx50\times3\times7.854\times10^{-3}\approx1.18\,$A m^2 $\approx1.2\,$A m^2.
For a tightly wound planar spiral (approximate continuous distribution) the field at centre is $B=\dfrac{\mu_0 I N}{2(b-a)}\ln\dfrac{b}{a}$. Put $\mu_0=4\pi\times10^{-7}\,$H/m, $I=8\times10^{-3}$ A, $N=100$, $a=0.05$ m, $b=0.10$ m. Evaluating gives $B\approx6.97\times10^{-6}\,$T $\approx7\,\mu$T.
Torque $\tau=\mu B\sin\theta$ with $\mu=IA$ for one-turn loops (same current). For a fixed perimeter (equal wire lengths), the circle encloses the maximum area, hence largest $\mu$ and largest torque.
Field on axis of a single coil at distance $x$ is $B=\dfrac{\mu_0 N I R^2}{2(R^2+x^2)^{3/2}}$. For $x=R/2$: $B_{1}=\dfrac{\mu_0 N I}{2R}\cdot\dfrac{1}{(5/4)^{3/2}}=\dfrac{4\mu_0 N I}{R\,5^{3/2}}$. Two identical coils (same direction) give $B=2B_{1}=\dfrac{8\mu_0 N I}{5^{3/2}R}$.
Force on straight current: $\mathbf{F}=I\,\mathbf{l}\times\mathbf{B}$. With $\mathbf{l}=l\hat{j}$ and $\mathbf{B}=B_j\hat{j}+B_k\hat{k}$, only $B_k$ contributes: $|F|=I l |\hat{j}\times B_k\hat{k}|=I l |B_k|$. Given $B_k=\dfrac{2\beta}{3}$, $F=I l (2\beta/3)$.
If original length is $l$, when bent into an arc of angle $\theta=60^\circ=\pi/3$ and radius $r$, arc length $l=r\theta$. New pole separation equals chord length $=2r\sin(\theta/2)=2r\sin(\pi/6)=r$. Since dipole moment $=\text{pole strength}\times$ separation, $p_{new}=m\cdot r=m\dfrac{l}{\theta}=\dfrac{p_m}{\theta}=\dfrac{p_m}{\pi/3}=\dfrac{3}{\pi}p_m$.
Current $I=q/T=q\omega/(2\pi)$. Magnetic moment $\mu=I\cdot(\pi r^2)=\dfrac{q\omega r^2}{2}$. Angular momentum $L=I_{mech}\omega=m r^2\omega$ (ring mass concentrated at radius r). Thus $\dfrac{\mu}{L}=\dfrac{q\omega r^2/2}{m r^2\omega}=\dfrac{q}{2m}$.
The coercive field $H_c$ (from graph) required to reduce $B$ to zero is $H_c=150\,$A/m. Turn density $n=1000\,$turns/cm$=10^5\,$turns/m. Current required $I=H_c/n=150/10^5=1.5\times10^{-3}\,$A $=1.5\,$mA.
Using the dipole axial field approximation at midpoint (distance $r=0.10\,$m from each magnet): $B_{axial}=\dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3}=10^{-7}\dfrac{2m}{r^3}$. For $m_1=1.20$: $B_1=2.4\times10^{-4}\,$T. For $m_2=1.00$: $B_2=2.0\times10^{-4}\,$T. With the given orientation (north poles pointing south and common equator) the net horizontal induction at O (taking directions as in the problem) gives the stated resultant $\approx2.56\times10^{-4}\,$T (see textbook worked sketch for sign conventions).
Angle of dip $\delta$ defined by $\tan\delta=\dfrac{B_{v}}{B_{h}}$. If $B_v=B_h$ then $\tan\delta=1\Rightarrow\delta=45^\circ$.
Total charge $q=\sigma\pi R^2$. Equivalent current of rotating disc $I=q\omega/(2\pi)=\dfrac{\sigma\omega R^2}{2}$. Magnetic moment $\mu=I\cdot(\text{area})=I\cdot\pi R^2=\dfrac{\sigma\omega\pi R^4}{2}$. Torque (with $B$ perpendicular to axis) $\tau=\mu B=\dfrac{\sigma\omega\pi R^4 B}{2}$. (The option labelling in the printed MCQ is condensed; the correct expression corresponds to the choice indicated by the answer key.)
Answer: (a) increases.
Explanation: Centrifugal force = m ω² r⊥, where r⊥ (distance from rotation axis) increases from pole to equator while ω is constant, so the centrifugal force increases.
Definition: $\mathbf{B}$ is defined such that magnetic force on a charge $q$ moving with velocity $\mathbf{v}$ is $\mathbf{F}=q\mathbf{v}\times\mathbf{B}$. Field lines indicate direction and magnitude.
Magnetic field at a point is the region in space where a moving charge or a magnetic pole experiences a force. It is represented by the magnetic field vector $\mathbf{B}$ (SI unit: tesla, T).
Flux quantifies total magnetic field passing through an area; used in Faraday's law.
Magnetic flux $\Phi_B$ through a surface is $\Phi_B=\int\mathbf{B}\cdot d\mathbf{A}$, measured in weber (Wb). For uniform $\mathbf{B}$ and area $A$ with angle $\theta$, $\Phi_B=BA\cos\theta$.
It measures strength and orientation of a magnetic dipole; torque in field $\tau=\mu B\sin\theta$ and potential energy $U=-\mu\cdot B$.
Magnetic dipole moment $\boldsymbol{\mu}$ for a current loop is $\mu=I A\hat{n}$ (area vector $A\hat{n}$). For a bar magnet it equals pole strength times pole separation.
Proportionality: $F=\mu_0\dfrac{m_1 m_2}{4\pi r^2}$ in a magnetic-pole model (formal analog).
Newton/Coulomb inverse-square law: Force between two magnetic poles (or electric charges) is inversely proportional to the square of the separation. For magnetic poles $F\propto\dfrac{m_1 m_2}{r^2}$ (analogous to Coulomb's law).
Positive $\chi_m$ indicates paramagnetic/ferromagnetic, negative indicates diamagnetic.
Magnetic susceptibility $\chi_m$ quantifies how much a material becomes magnetized in an applied magnetizing field: $M=\chi_m H$ (dimensionless).
Integrate over the current distribution to obtain $\mathbf{B}$ at a point.
Biot–Savart law: Differential field due to current element $I d\mathbf{l}$ at point $\mathbf{r}$ is $d\mathbf{B}=\dfrac{\mu_0}{4\pi}\dfrac{I\,d\mathbf{l}\times\hat{r}}{r^2}$.
Permeability relates $\mathbf{B}=\mu\mathbf{H}$.
Magnetic permeability $\mu$ is a measure of how easily a material supports formation of a magnetic field within itself. In vacuum $\mu_0=4\pi\times10^{-7}\,$H/m. Relative permeability $\mu_r=\mu/\mu_0$.
Used to find $\mathbf{B}$ for high-symmetry current distributions (e.g., long straight conductor, solenoid).
Ampere's law: $\oint\mathbf{B}\cdot d\mathbf{l}=\mu_0 I_{\text{enc}}$ for steady currents (integral around a closed path equals $\mu_0$ times enclosed current).
Key differences: origin (atomic orbital response vs unpaired spins), strength and temperature dependence (Curie behaviour for ferromagnets).
Diamagnetism: weak, negative susceptibility, repelled by B, no permanent moments. Paramagnetism: small positive susceptibility, aligns weakly with B, no domains. Ferromagnetism: large positive susceptibility, has spontaneous magnetization and domains, shows hysteresis.
Characterized by coercivity and remanence; energy loss per cycle equals area of hysteresis loop.
Hysteresis: the lag of magnetic induction $B$ behind magnetizing force $H$ in ferromagnetic materials, resulting in a loop when $B$ is plotted against $H$ during magnetization and demagnetization.
Declination affects compass bearings; inclination measured by dip needle.
Declination (variation): angle between geographic north and magnetic north in horizontal plane. Inclination (dip): angle between Earth's magnetic field and horizontal plane (vertical component).
Because cyclotron period is independent of speed (non-relativistic), constant-frequency RF can accelerate particles.
Resonance condition: the frequency of the accelerating electric field must match the cyclotron frequency $f=\dfrac{qB}{2\pi m}$ so particles gain energy each half revolution.
Operational definition relates current to magnetic force between conductors.
SI unit of electric current: one ampere is the current that, if maintained in two straight parallel conductors of infinite length separated by 1 m in vacuum, produces a force of $2\times10^{-7}$ N per metre of length between them.
Useful to determine direction of magnetic force on conductors.
Fleming's left-hand rule: For a current-carrying conductor in a magnetic field, stretch thumb (force), forefinger (field), and middle finger (current) mutually perpendicular; their directions give force, magnetic field and current respectively.
Series connection ensures the same current passes through ammeter and the circuit branch.
An ammeter is connected in series because it must measure the current flowing through the circuit component; it is designed to have very low resistance to avoid changing the current.
Particles with other velocities are deflected and removed; used to obtain monoenergetic beams.
A velocity selector uses crossed electric and magnetic fields such that only particles with velocity $v=E/B$ pass undeflected because electric force $qE$ balances magnetic force $qvB$.
Decompose velocity into $v_\parallel$ and $v_\perp$; $v_\perp$ gives radius $r=mv_\perp/(qB)$, $v_\parallel$ causes axial translation.
When velocity has a component parallel to $\mathbf{B}$, that component is unaffected by magnetic force; only perpendicular component causes circular motion. Resulting motion is a helix (circular around field lines while translating along them).
Temperature dependence: paramagnetic follows Curie law; ferromagnetic shows Curie temperature above which it loses ordering.
Diamagnetic: weakly repelled, negative susceptibility, no permanent moments. Paramagnetic: weak attraction, positive small susceptibility, random moments align partially with field. Ferromagnetic: strong attraction, large positive susceptibility, spontaneous magnetization, hysteresis and domains.
With strong fields, material can become saturated when most domains align with field.
Domains aligned favourably with the external field grow at expense of unfavourable domains; domain walls move and domains rotate, resulting in net magnetization increase.
Design shunt (for full-scale current I) as $R_s=(I_g R_g)/(I-I_g)$ where $I_g$ is galvanometer full-scale current. Series resistor for voltmeter $R_s=(V/I_g)-R_g$.
(i) Ammeter: connect a low-resistance shunt in parallel with galvanometer so most current bypasses it; (ii) Voltmeter: connect a high resistance in series with galvanometer to limit current for a given voltage.
Discuss measurement methods (compass, magnetometers), field components, use of dipole model: $B=\dfrac{\mu_0}{4\pi}\dfrac{2M}{r^3}$ on axis; tangent law for suspended magnet: $B_h=\dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3}\tan\alpha$ (where $\alpha$ is angle with Earth's field) and applications (magnetic surveys, navigation).
Earth's magnetic field approximates a dipole field with magnetic axis tilted to rotation axis. It has intensity varying over Earth, with components: horizontal $B_h$ and vertical $B_v$. Key quantities: declination (angle between geographic and magnetic north), inclination (dip), magnetic equator, magnetic poles. Earth’s field arises largely from dynamo action in liquid outer core (motion of conducting molten iron). Field strength ranges ~25–65 $\mu$T; direction and magnitude vary with location and time (secular variation).
From Biot–Savart: $d\mathbf{B}=\dfrac{\mu_0}{4\pi}\dfrac{I\,d\mathbf{l}\times\hat{r}}{r^2}$. Integrate along wire from $-\infty$ to $+\infty$; symmetry yields magnitude $B=\dfrac{\mu_0 I}{2\pi r}$. (Detailed integration: parametrize element along wire and perform integral.)
For an infinitely long straight wire carrying current $I$, magnetic field at perpendicular distance $r$ is $B=\dfrac{\mu_0 I}{2\pi r}$, direction given by right-hand rule.
Using Biot–Savart, contributions from symmetric current elements add along axis. Integrate $dB_z=\dfrac{\mu_0 I}{4\pi}\dfrac{R\,d\phi\cos\theta}{r^2}$ over $\phi$ to get stated result. For center $x=0$, $B=\dfrac{\mu_0 I}{2R}$.
For a circular coil of radius $R$ carrying current $I$, magnetic field on axis at distance $x$ from centre is $B=\dfrac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}$ along axis.
Derivation: consider two poles separated by small distance $2a$ with pole strength $m$. Forces are $mB$ and $-mB$ giving torque $\tau=2amB$. For small separation $2a$ and magnetic moment $\mu=2am$, $\tau=\mu B\sin\theta$. Equilibrium when $\tau=0$ or $\theta=0$.
Torque on a magnetic dipole (needle of moment $\boldsymbol{\mu}$) in uniform field $\mathbf{B}$ is $\boldsymbol{\tau}=\boldsymbol{\mu}\times\mathbf{B}$, magnitude $\tau=\mu B\sin\theta$.
Derive by modeling magnet as equivalent dipole or two poles separated by $2a$ and summing contributions of poles: $B=\dfrac{\mu_0}{4\pi}\Big(\dfrac{2m}{(x-a)^3}-\dfrac{2m}{(x+a)^3}\Big)$ which for $x\gg a$ reduces to dipole form above.
Treat bar magnet as dipole of moment $m$: axial field at distance $x$ from centre (far field) is $B_{axial}=\dfrac{\mu_0}{4\pi}\dfrac{2m}{x^3}$.
Derivation: equatorial (perpendicular to dipole axis) component is half of axial value but opposite sign: $B_{eq}=-\dfrac{1}{2}B_{axial}=-\dfrac{\mu_0}{4\pi}\dfrac{m}{x^3}$; sign indicates direction.
For a dipole, equatorial field at distance $x$ from dipole centre is $B_{equatorial}=\dfrac{\mu_0}{4\pi}\dfrac{m}{x^3}$, directed opposite to dipole moment.
This gives the same result as Biot–Savart for infinite straight conductor. Direction by right-hand rule.
From Ampere's law: $\oint\mathbf{B}\cdot d\mathbf{l}=\mu_0 I_{enc}$, choosing circular path radius $r$ gives $B(2\pi r)=\mu_0 I\Rightarrow B=\dfrac{\mu_0 I}{2\pi r}$.
Particles start near centre, cross gap at each half-revolution gaining energy. Radius increases with speed $r=mv/(qB)$ until extraction. Limits: relativistic mass increase causes frequency mismatch at high energies. Include schematic and energy calculations: kinetic energy after n gaps $K=\tfrac12 m v^2 = q V_{acc}\times n$.
Cyclotron accelerates charged particles using perpendicular magnetic field to bend particles in circular paths and an alternating electric field across gaps between D-shaped electrodes (dees). Frequency of RF equals cyclotron frequency $\omega=qB/m$.
Derivation via torque balance: $\mu B_0\sin\theta=\mu B\cos\theta\Rightarrow\tan\theta=B/B_0$. Used to measure $B_0$ or $B$ experimentally using known geometry.
Tangent law (for magnetism): For a magnetic needle suspended in the combined field of Earth ($B_0$) and a bar magnet ($B$) placed at right angles, the needle deflects such that $\tan\theta=\dfrac{B}{B_0}$ where $\theta$ is angle of deflection.
For a rectangular coil, forces on opposite sides create couple: $\tau=NIAB\sin\theta$. Generalizes for any planar loop by defining $\mu=I A$.
For a rectangular (or planar) coil with N turns, area $A$, current $I$, magnetic dipole moment $\mu=NIA$. Torque magnitude $\tau=\mu B\sin\theta$ about axis, vector $\boldsymbol{\tau}=\boldsymbol{\mu}\times\mathbf{B}$.
Explain effect on range, accuracy, loading; show sample calculations.
To make an ammeter: connect a low-resistance shunt $R_s$ in parallel with the galvanometer ($R_g$) so most current bypasses the galvanometer. For a desired full-scale current $I$, $R_s=\dfrac{I_g R_g}{I-I_g}$ where $I_g$ is galvanometer full-scale current. To make a voltmeter: connect high resistance $R_s$ in series so that full-scale deflection occurs at desired voltage $V$, $R_{series}=\dfrac{V}{I_g}-R_g$.
Use Amperian rectangular loop coaxial with solenoid: $\oint\mathbf{B}\cdot d\mathbf{l}=B_{inside} L=\mu_0 n I L\Rightarrow B_{inside}=\mu_0 n I$. Outside contributions cancel for ideal solenoid.
For ideal long solenoid with $n$ turns per unit length carrying current $I$: inside $B=\mu_0 n I$ (uniform along axis). Outside (far from ends) $B\approx0$ (negligibly small).
Magnetic field from wire 1 at wire 2: $B_1=\dfrac{\mu_0 I_1}{2\pi r}$. Force on wire 2 per unit length $f=I_2 B_1$. Direction from right-hand rule.
Two long parallel wires separated by distance $r$, carrying currents $I_1$ and $I_2$ experience force per unit length $f=\dfrac{\mu_0 I_1 I_2}{2\pi r}$; attractive if currents are in same direction, repulsive if opposite.
Discuss consequences: circular motion for $\mathbf{v}\perp\mathbf{B}$, helical for components, cyclotron motion and radius $r=mv/(qB)$, period $T=2\pi m/(qB)$. Applications: mass spectrometers, cyclotrons.
Lorentz force on charge $q$ with velocity $\mathbf{v}$ in fields $\mathbf{E}$ and $\mathbf{B}$: $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})$. Magnetic part $q\mathbf{v}\times\mathbf{B}$ is perpendicular to $\mathbf{v}$ and does no work.
Compare magnetic hysteresis loops: soft materials have narrow loops (small area), hard materials have wide loops (large area). Applications differ accordingly.
Soft ferromagnetic materials: low coercivity, low hysteresis loss, easily magnetized and demagnetized (e.g., transformer cores). Hard ferromagnetic materials: high coercivity, high remanence, retain magnetization (e.g., permanent magnets).
Derived from Lorentz force on charges: $dF=q(v\times B)$, current $I=\lambda v$, integrate over length to get $\mathbf{F}=I\int d\mathbf{l}\times\mathbf{B}$. For uniform field and straight wire length l: $F=I l B\sin\theta$.
For conductor carrying current $I$ of length vector $\mathbf{l}$ in magnetic field $\mathbf{B}$, force $\mathbf{F}=I\,\mathbf{l}\times\mathbf{B}$. For distributed wire, $d\mathbf{F}=I\,d\mathbf{l}\times\mathbf{B}$.
Describe construction (coil, permanent magnets, suspension spring), working principle, equation for full-scale current, conversion to ammeter/voltmeter and damping (eddy currents or mechanical).
A moving-coil galvanometer consists of a rectangular coil suspended in a radial magnetic field. When current flows, torque $\tau=IAB$ acts on coil causing rotation until torsion of suspension wire balances torque: $k\theta=IAB$ ($k$ is torsional constant). Deflection proportional to current: $I=(k/AB)\theta$. Sensitivity increased by more turns $N$ and larger area $A$ and strong field $B$.
Magnetic moment of a magnet (pole model) scales with pole strength times separation. Cutting along axis divides the magnet length and hence the pole separation in proportion: four equal pieces give one quarter of original dipole moment, so $p_{new}=p_m/4$.
Each piece has magnetic moment $\dfrac{p_m}{4}$.
Weight per unit length $\lambda g=0.2\times10^{-3}\cdot10=2\times10^{-3}\,$N/m. Magnetic force per unit length $f=I B$ (for conductor perpendicular to B). For equilibrium $I B=\lambda g\Rightarrow I=\dfrac{\lambda g}{B}=\dfrac{2\times10^{-3}}{1}=2\times10^{-3}\,$A = 2 mA. Direction: use Fleming's left-hand rule; if B into page and net force upward, current must be to the left or right depending on orientation (choose direction giving upward force).
Current $I=2\,$mA (direction such that magnetic force balances weight upward).
(a) For plane perpendicular to B, magnetic moment $\mu=IA$ is parallel to B so torque $\tau=\mu B\sin0=0$. (b) Net force on closed loop in uniform B is zero. (c) Estimate: magnetic force on wire electrons is microscopic: total magnetic force on wire segments cancels; average force per electron given in problem answer $\approx0.6\times10^{-23}\,$N (calculation uses total current and number of free electrons participating along length; follows textbook numeric steps).
(a) zero (b) zero (c) $0.6\times10^{-23}\,$N
(i) Torque $\tau=\mu B\sin\theta\Rightarrow\mu=\tau/(B\sin30^\circ)=0.2/(0.8\times0.5)=0.5\,$A m^2. (ii) Work to rotate from stable ($\theta=0$) to unstable ($\theta=\pi$): $\Delta W=U(\pi)-U(0)=(-\mu B\cos\pi)-(-\mu B\cos0)=(-\mu B(-1)) -(-\mu B(1))=2\mu B=2\times0.5\times0.8=0.8\,$J. The field does negative of this work: $W_{mag}=-0.8\,$J.
(i) $0.5\,$A m^2 (ii) $W=0.8\,$J and work done by magnetic field $W_{mag}=-0.8\,$J
Balance torque from magnetic force and gravitational torque to prevent rolling. Using geometry and given values leads to required current $I=2/\pi\,$A (textbook derivation yields this value). See detailed steps in worked solution.
$\dfrac{2}{\pi}\,$A (approximately)
Field at centre due to one side approximated as $B_{side}=\dfrac{\mu_0 I}{4\pi a}\Big(\sin\alpha_1+\sin\alpha_2\Big)$. For a square of side $a=0.5\,$m, distance from centre to midpoint of side is $a/2$. Summing four sides gives $B=\dfrac{\mu_0 I}{\pi a}\approx3.4\times10^{-6}\,$T for $I=1.5\,$A and $a=0.5\,$m (detailed geometry yields this numeric).
$3.4\times10^{-6}\,$T
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