Since $1/(\mu_0\varepsilon_0)=c^2$, and speed squared has dimension $[L^2T^{-2}]$, the required dimension is $[L^{2}T^{-2}]$.
In vacuum $E_0=cB_0$. With $c\approx3\times10^8\,$m/s and $B_0=3\times10^{-6}\,$T, $E_0\approx3\times10^8\times3\times10^{-6}=9\times10^2\,$V/m.
Infrared radiation can penetrate some amounts of fog and is used in thermal imaging and night vision systems.
Electromagnetic waves are transverse (E and B perpendicular to direction of propagation), so the statement 'longitudinal' is false.
Wavelength $\lambda=c/f$. With $f=300\,$MHz $=3\times10^8\,$Hz and $c\approx3\times10^8\,$m/s, $\lambda\approx1\,$m.
For propagation along $-\hat{i}$, we need $\mathbf{E}\times\mathbf{B}=-\hat{i}$. With $\mathbf{E}=\hat{k}$ and $\mathbf{B}=\hat{j}$, $\hat{k}\times\hat{j}=-\hat{i}$, so option B is consistent.
E_{rms}=3 V/m so E_0=\sqrt{2}E_{rms}=4.243 V/m. Peak magnetic field $B_0=E_0/c\approx4.243/(3\times10^8)=1.414\times10^{-8}\,$T.
For propagation along +x, we need $\mathbf{E}\times\mathbf{B}=\hat{i}$. With $\mathbf{E}=\hat{j}$ and $\mathbf{B}=\hat{k}$, $\hat{j}\times\hat{k}=\hat{i}$, so $\mathbf{B}$ is +z.
Gauss's law for magnetism states $\oint_s\mathbf{B}\cdot d\mathbf{A}=0$, expressing no magnetic monopoles. If monopoles exist this equation would have a nonzero right-hand side (magnetic charge enclosed).
Fraunhofer lines are dark lines in the solar spectrum produced when cooler gases absorb specific wavelengths from the continuous spectrum, i.e. an absorption-line spectrum.
Gamma rays are high-energy photons and hence electromagnetic. Alpha and beta rays are charged particle radiation.
Changing (accelerating) charges produce time-varying fields that radiate electromagnetic waves. A uniformly moving or stationary charge does not radiate.
Here wave number $k=10^6\,$m^{-1}. For EM waves $\omega=ck\approx(3\times10^8)(10^6)=3\times10^{14}\,$rad/s.
In vacuum all electromagnetic waves travel at the same speed $c$, independent of frequency; hence option D is not true.
For a plane electromagnetic wave in free space, the electric and magnetic fields oscillate in phase and are perpendicular to each other and to the direction of propagation.
Maxwell added displacement current density $\mathbf{J}_d=\varepsilon_0\dfrac{\partial\mathbf{E}}{\partial t}$. In integral form Ampère–Maxwell law becomes $\displaystyle\oint\mathbf{B}\cdot d\mathbf{l}=\mu_0 I_{\text{enc}}+\mu_0\varepsilon_0\dfrac{d}{dt}\Phi_E$. This explains magnetic effects in regions with changing electric flux.
Displacement current is the term $\varepsilon_0\dfrac{\partial \mathbf{E}}{\partial t}$ (or $\varepsilon_0\dfrac{d\Phi_E}{dt}$ in integral form) introduced by Maxwell to account for the rate of change of electric field in regions (e.g. between capacitor plates) where there is no conduction current; it contributes to the magnetic field exactly as a real current does.
They consist of mutually perpendicular electric and magnetic fields in phase; examples include radio waves, light, X-rays, and gamma rays.
Electromagnetic waves are self-sustaining transverse waves of oscillating electric and magnetic fields that propagate through space carrying energy and momentum; they satisfy Maxwell's equations and travel at speed $c=1/\sqrt{\mu_0\varepsilon_0}$ in vacuum.
The extra term $\mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$ is the displacement current term which restores consistency with charge conservation.
Ampère–Maxwell law (integral form): $\displaystyle\oint_{\ell}\mathbf{B}\cdot d\mathbf{l}=\mu_0 I_{\rm enc}+\mu_0\varepsilon_0\dfrac{d}{dt}\Phi_E$, where $\Phi_E=\int_{S}\mathbf{E}\cdot d\mathbf{A}$ is electric flux through surface bounded by $\ell$.
This is a fundamental Maxwell equation indicating net magnetic flux through any closed surface is zero. If magnetic monopoles existed the right-hand side would be magnetic charge enclosed.
Gauss's law for magnetism states $\displaystyle\oint_S\mathbf{B}\cdot d\mathbf{A}=0$ for any closed surface $S$, implying there are no isolated magnetic charges (monopoles) and magnetic field lines are continuous closed loops.
IR detects heat and is used in sensors; microwaves heat dielectric materials and are used in radar links; UV kills bacteria and is used to sterilize water/surfaces.
(i) IR: thermal imaging (night vision), remote control/communication. (ii) Microwaves: microwave ovens (heating), radar and long-distance communication. (iii) UV: sterilization/disinfection, photochemical processes and fluorescent lamps.
Each element produces characteristic spectral lines; comparing solar absorption line wavelengths with laboratory emission/absorption lines reveals the Sun's elemental composition.
Fraunhofer lines are dark absorption lines in the solar spectrum produced when specific wavelengths are absorbed by cooler gases in the Sun's atmosphere. By matching the wavelengths of these absorption lines to known atomic spectra, the elements present in the Sun's atmosphere can be identified.
The displacement current term explains magnetic fields in regions without conduction current (e.g. between capacitor plates) and allows Maxwell's equations to predict electromagnetic waves.
Ampère–Maxwell law generalizes Ampère's law by adding the displacement current term: $\oint\mathbf{B}\cdot d\mathbf{l}=\mu_0 I_{\rm enc}+\mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$. This term accounts for time-varying electric fields and ensures continuity of current and consistency with charge conservation.
Mechanical waves need particles of a medium to transfer energy; EM waves transmit energy via fields themselves, independent of matter.
Electromagnetic waves are non-mechanical because they do not require a material medium to propagate; they are oscillations of electric and magnetic fields in space and can travel through vacuum.
These four integral relations describe how charges produce electric fields, absence of magnetic monopoles, how time-varying magnetic flux induces electric fields, and how currents and changing electric flux produce magnetic fields.
Maxwell's equations (integral form):
- Gauss's law (electric): $\displaystyle\oint_S\mathbf{E}\cdot d\mathbf{A}=\dfrac{Q_{\rm enc}}{\varepsilon_0}$.
- Gauss's law (magnetic): $\displaystyle\oint_S\mathbf{B}\cdot d\mathbf{A}=0$.
- Faraday's law: $\displaystyle\oint_{\ell}\mathbf{E}\cdot d\mathbf{l}=-\dfrac{d}{dt}\Phi_B$, where $\Phi_B=\int_S\mathbf{B}\cdot d\mathbf{A}$.
- Ampère–Maxwell law: $\displaystyle\oint_{\ell}\mathbf{B}\cdot d\mathbf{l}=\mu_0 I_{\rm enc}+\mu_0\varepsilon_0\dfrac{d}{dt}\Phi_E$, where $\Phi_E=\int_S\mathbf{E}\cdot d\mathbf{A}$.
Each band has characteristic frequencies, sources, interactions with matter and practical applications; e.g., visible light is used in optics, X-rays for imaging, microwaves for heating and communication, radio waves for broadcasting and telecommunication.
(a) Microwaves: Frequency range ~300 MHz to 300 GHz. Used in cooking (dielectric heating), radar, satellite communication, and microwave links. They penetrate clouds and are useful in remote sensing.
(b) X-rays: High-energy electromagnetic radiation (approx. 10^{16}–10^{19} Hz). Strongly penetrating, used in medical imaging (radiography), crystallography, and industrial inspection. Produced by accelerated electrons hitting targets or by inner-shell electronic transitions.
(c) Radio waves: Lowest-frequency part of EM spectrum (~kHz to GHz). Used for AM/FM broadcasting, TV, two-way communication, and navigation. Long wavelengths allow long-distance propagation via ionospheric reflection.
(d) Visible spectrum: Wavelengths ~400–700 nm corresponding to light perceptible by human eye. Contains colours from violet to red; important for vision, photosynthesis, and optical instruments.
Hertz's experiments in 1887–88 confirmed Maxwell's theory by producing and detecting radio waves, measuring their wavelength and speed, and demonstrating wave properties (reflection, refraction, polarization). This established that light is an electromagnetic wave.
Hertz demonstrated the existence of electromagnetic waves predicted by Maxwell. He used an oscillating spark-gap transmitter (an RLC oscillator producing oscillating charges) to generate radio-frequency EM waves and detected them with a receiving loop that produced sparks when in resonance. He showed that these waves underwent reflection, refraction, polarization, interference and had the same speed as light.
Key points: generation by accelerated charges, detection by induced oscillations in a resonant circuit, measurement of wavelength via standing wave patterns, verification that EM waves behave like light.
The displacement current term acts like a real current in producing magnetic fields, ensures continuity equation $\nabla\cdot\mathbf{J}+\partial\rho/\partial t=0$ is satisfied, and leads to wave equations for fields allowing propagation of EM waves.
Maxwell added the displacement current term to Ampère's law to resolve an inconsistency with charge conservation. Original Ampère's law $\oint\mathbf{B}\cdot d\mathbf{l}=\mu_0 I_{\rm enc}$ fails for cases like a charging capacitor: different surfaces bounded by same loop give different enclosed currents. Maxwell introduced displacement current density $\mathbf{J}_d=\varepsilon_0\partial\mathbf{E}/\partial t$. The modified law becomes $\oint\mathbf{B}\cdot d\mathbf{l}=\mu_0 I_{\rm enc}+\mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$, restoring consistency and symmetry between electric and magnetic fields.
Without the displacement current term, Maxwell's theory would not predict self-propagating EM waves. The correction enabled predictions later experimentally confirmed, forming the basis of classical electrodynamics and optics.
Maxwell's correction (displacement current) is important because:
- It resolves the inconsistency in Ampère's law for time-varying fields (e.g. between capacitor plates).
- It enforces charge conservation and the continuity equation.
- It symmetrizes Maxwell's equations between electric and magnetic fields.
- Crucially, it leads to the prediction of electromagnetic waves: combining curl equations yields wave equations with speed $c=1/\sqrt{\mu_0\varepsilon_0}$. Thus electromagnetic radiation and light are unified.
These properties follow from Maxwell's equations and the wave solutions for E and B fields.
Properties of electromagnetic waves:
- Transverse: E and B are perpendicular to the direction of propagation and to each other.
- Mutually sustaining: time-varying E produces B and vice versa.
- Travel in vacuum at speed $c=1/\sqrt{\mu_0\varepsilon_0}$.
- Carry energy and momentum; energy density $u=\tfrac{1}{2}\varepsilon_0 E^2+\tfrac{1}{2\mu_0}B^2$ and Poynting vector $\mathbf{S}=\tfrac{1}{\mu_0}\mathbf{E}\times\mathbf{B}$ gives energy flux.
- In free space E and B are in phase and related by $E_0=cB_0$.
- Can be polarized, reflected, refracted, diffracted and interfered.
- Do not require a material medium (non-mechanical).
- Obey superposition principle.
A localized oscillating charge (e.g. dipole antenna) radiates power; the instantaneous power radiated by an accelerating charge is given by Larmor formula $P=\dfrac{q^2a^2}{6\pi\varepsilon_0 c^3}$ (non-relativistic).
Electromagnetic waves are produced by time-varying currents and accelerating charges. An accelerating charge produces changing electric and magnetic fields that propagate outward as radiation. Common sources: oscillating charges in antennas (radio waves), transitions of electrons between energy levels in atoms (visible, UV, X-rays), and bremsstrahlung from decelerating charged particles (X-rays).
Atomic emission lines are used for element identification; continuous spectra depend on temperature (Planck's law); molecular bands appear broadened due to many closely spaced transitions.
Types of emission spectra:
- Continuous spectrum: emitted by hot dense objects (blackbody) showing a continuous spread of wavelengths.
- Line (discrete) emission spectrum: produced by atoms or ions when electrons transition from higher to lower energy levels; appears as discrete bright lines at characteristic wavelengths (e.g. atomic emission lines).
- Band emission spectrum: arises from molecular transitions where vibrational and rotational sub-levels produce groups (bands) of closely spaced lines; typical of molecules and some solids.
Each type depends on the nature of the emitter—atoms produce line spectra, molecules band spectra, and thermal sources continuous spectra.
By comparing absorption lines/bands with known atomic or molecular spectra, one can identify substances and infer conditions in stellar atmospheres or laboratory samples.
Types of absorption spectra:
- Continuous absorption: a continuous range of wavelengths absorbed (e.g. broad attenuation by a continuum absorber).
- Line (discrete) absorption spectrum: produced when a cooler low-density gas absorbs specific wavelengths from a continuous background source, producing dark lines at characteristic wavelengths corresponding to electronic transitions (Fraunhofer lines in the solar spectrum).
- Band absorption spectrum: molecules absorb in bands due to vibrational/rotational transitions, producing groups of dark bands.
Absorption spectra provide information about the composition and physical conditions (temperature, pressure) of the absorbing medium.
When the conduction current in the connecting wire is 5 A charging the capacitor, the rate of change of electric flux between the plates produces an equal displacement current through any surface between plates. Thus $I_d=I_{\rm conduction}=5\,$A (by continuity of current).
Displacement current $I_d=5\,$A
Oscillation angular frequency $\omega=1/\sqrt{LC}$. With $L=1\times10^{-6}\,$H and $C=1\times10^{-6}\,$F, $\omega=1/\sqrt{10^{-12}}=10^{6}\,$rad/s. Frequency $f=\omega/(2\pi)=10^6/(2\pi)\approx1.592\times10^{5}\,$Hz. Wavelength $\lambda=c/f\approx(3\times10^8)/(1.592\times10^{5})\approx1.884\times10^{3}\,$m.
Approximately $1.884\times10^{3}\,$m
Energy of pulse $E=Pt=(60\times10^{-3})(10^{-6})=6.0\times10^{-8}\,$J. Photons carry momentum $p=E/c$. Thus $p=(6.0\times10^{-8})/(3\times10^8)=2.0\times10^{-16}\,$kg m s^{-1}.
Final momentum $p=2.0\times10^{-16}\,$kg m s^{-1}
Given frequency $f=10^{10}$ Hz, $\lambda=c/f=(3\times10^8)/(10^{10})=3\times10^{-2}\,$m. For a wave traveling in +x with $\mathbf{B}$ along +y, $\mathbf{E}$ must be along +z with $E_0=cB_0=3\times10^3\,$V/m. So $\mathbf{E}(x,t)=3\times10^3\sin\left(2\pi\times10^{10}t -\dfrac{2\pi}{\lambda}x\right)\hat{z}$ or equivalently $3\times10^3\sin(kx-\omega t)\,\hat{z}$ with $\omega=2\pi\times10^{10}$ and $k=2\pi/\lambda$ (sign convention depending on propagation phase).
Wavelength $\lambda=3\times10^{-2}\,$m. Electric field: $\mathbf{E}(x,t)=E_0\sin(kx-\omega t)\,\hat{z}$ with $E_0=cB_0\approx3\times10^8\times10^{-5}=3\times10^3\,$V/m, $\omega=2\pi\times10^{10}\,$s^{-1}, $k=\omega/c$.
In a medium $v=1/\sqrt{\mu\varepsilon}=c/\sqrt{\mu_r\varepsilon_r}$. With $\mu_r=1.0$, $\varepsilon_r=2.25$, $v=c/\sqrt{2.25}=c/1.5=(3\times10^8)/1.5=2.0\times10^8\,$m/s.
Speed $v=2.0\times10^{8}\,$m s^{-1}
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