Vardaan Learning Institute

Ans: When evaluating $\oint \vec{B} \cdot d\vec{l} = \mu_0 I$ around a charging capacitor, Maxwell found that taking a flat surface intersecting the wire gave a current $I$, but taking a balloon-like surface passing strictly through the vacuum gap between the plates gave $I = 0$. This resulted in two different magnetic fields for the exact same loop, a glaring logical inconsistency.

Ans: Total Current $I = I_c + I_d$, where $I_c$ is the conduction current in the wires, and $I_d = \epsilon_0 \frac{d\Phi_E}{dt}$ is the displacement current in the gap.

Ans: Electric field between plates $E = \frac{\sigma}{\epsilon_0} = \frac{q}{A\epsilon_0}$. Electric flux $\Phi_E = E \times A = \frac{q}{\epsilon_0}$.
Displacement current $I_d = \epsilon_0 \frac{d\Phi_E}{dt} = \epsilon_0 \frac{d}{dt}\left(\frac{q}{\epsilon_0}\right) = \frac{dq}{dt}$.
Since $dq/dt$ is exactly the conduction current ($I_c$) flowing in the wires, $I_d = I_c$.

Ans: We proved $I_d = I_c = dq/dt$. Since $q = CV$, $I_d = \frac{d}{dt}(CV) = C \frac{dV}{dt}$.
$I_d = (2.0 \times 10^{-6}) \times 50 = 100 \times 10^{-6}\text{ A} = 10^{-4}\text{ A} = 0.1\text{ mA}$.

Ans: Yes. Maxwell proved that a time-varying electric field (displacement current) is physically equivalent to a conduction current in its ability to generate a surrounding magnetic field.

Ans: The fundamental origin is a **time-varying electric field** (or changing electric flux) occurring in the region between the capacitor plates.

Ans: In a steady DC state, the capacitor is fully charged, and the voltage and electric field $E$ between the plates become perfectly constant over time. Since $E$ is constant, the rate of change of electric flux $d\Phi_E/dt = 0$, making $I_d = 0$.

Ans: No, it does NOT represent the physical flow of electrons or any mass. Conduction current is the actual drift of charged particles (like electrons in a wire). Displacement current is merely a mathematical equivalent current arising from a changing electric field in a vacuum or dielectric.

Ans: Yes. A radio wave is an electromagnetic wave, which inherently consists of rapidly oscillating (time-varying) electric and magnetic fields. This changing electric field in the vacuum constitutes a continuous displacement current.

Ans: Rate of change of flux $\frac{d\Phi_E}{dt} = \frac{200 - 0}{2 \times 10^{-6}} = 100 \times 10^6 = 10^8\text{ V}\cdot\text{m/s}$.
$I_d = \epsilon_0 \frac{d\Phi_E}{dt} = (8.85 \times 10^{-12}) \times 10^8 = 8.85 \times 10^{-4}\text{ A}$.

Ans: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_0}$. It signifies that electric charges are the fundamental sources (or sinks) of electric fields, and that isolated positive or negative electric charges (monopoles) can exist in nature.

Ans: $\oint \vec{B} \cdot d\vec{A} = 0$. It profoundly implies that magnetic monopoles (isolated North or South poles) absolutely do not exist in the universe. Magnetic field lines always form continuous, closed loops.

Ans: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$ (The line integral of electric field around a closed loop equals the negative rate of change of magnetic flux through the loop).

Ans: The Ampere-Maxwell Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_c + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$. The second term explicitly shows that a changing electric flux ($\Phi_E$) generates a magnetic field ($\vec{B}$).

Ans: A **time-varying magnetic field** (or changing magnetic flux $\Phi_B$). This induced electric field is non-conservative, unlike electrostatic fields created by stationary charges.

Ans: Maxwell's second equation (Gauss's law for magnetism). It would change from $\oint \vec{B} \cdot d\vec{A} = 0$ to $\oint \vec{B} \cdot d\vec{A} = \mu_0 q_m$ (where $q_m$ is the magnetic monopole charge).

Ans: Faraday's law states a changing $\vec{B}$ field creates an $\vec{E}$ field. The Ampere-Maxwell law states a changing $\vec{E}$ field creates a $\vec{B}$ field. In free space, this creates a perfect, symmetric feedback loop where oscillating E and B fields continuously regenerate each other, propagating as an EM wave.

Ans: In a pure vacuum far from any sources, there are no charges ($Q=0$) and no conduction currents ($I_c=0$). The equations simplify to perfect symmetry between E and B, relying solely on displacement current and induction.

Ans: Both. Maxwell's equations are universally valid fundamental laws of classical physics, applying flawlessly to both large-scale macroscopic phenomena (like antennas) and microscopic atomic domains.

Ans: The Lorentz Force Law: $\vec{F} = q\vec{E} + q(\vec{v} \times \vec{B})$. Maxwell's equations describe how fields are created; Lorentz force describes how fields act on matter.

Ans: It means that the oscillating electric field ($\vec{E}$) and magnetic field ($\vec{B}$) vectors are both perpendicular to the direction in which the wave is traveling (propagating), as well as being perpendicular to each other.

Ans: They form a mutually perpendicular (orthogonal) triad. If $\vec{E}$ is along the X-axis and $\vec{B}$ is along the Y-axis, then the wave velocity $\vec{c}$ must be along the Z-axis.

Ans: Exactly zero ($0^\circ$). The electric and magnetic fields oscillate perfectly in phase, reaching their maxima and zero minima at the exact same instant and position in space.

Ans: $v = \frac{1}{\sqrt{\mu \epsilon}} = \frac{1}{\sqrt{\mu_0 \mu_r \epsilon_0 \epsilon_r}} = \frac{c}{\sqrt{\mu_r \epsilon_r}}$, where $c = 1/\sqrt{\mu_0 \epsilon_0}$ is the speed of light in vacuum.

Ans: $v = \frac{c}{n} = \frac{3.0 \times 10^8}{1.50} = 2.0 \times 10^8\text{ m/s}$.

Ans: $E_0 = c \times B_0 = (3 \times 10^8) \times (5 \times 10^{-7}) = 15 \times 10^1 = 150\text{ V/m}$.

Ans: Total energy density is the sum of energy stored in the electric and magnetic fields per unit volume. The energy is shared exactly equally. Therefore, the ratio $u_E : u_B$ is $1:1$.

Ans: Since $u_E = u_B$, total $u = 2 u_E$. We know $u_E = \frac{1}{2} \epsilon_0 E_{rms}^2$. Therefore, total average energy density $u = \epsilon_0 E_{rms}^2$.

Ans: Radiation pressure is the mechanical pressure (force per unit area) exerted by an EM wave when it strikes a surface. For a completely absorbing surface, Pressure $P = \frac{I}{c}$ (Intensity divided by speed of light).

Ans: For a fully *reflecting* surface, the wave bounces back, resulting in double the momentum change. Momentum transferred $p = \frac{2U}{c}$. (If it were absorbing, it would just be $U/c$).

Ans: $c = f \lambda$ (or $c = \nu \lambda$). Speed equals frequency multiplied by wavelength.

Ans: No. Electromagnetic waves are uncharged. Since they carry no net electric charge, they experience zero Lorentz force and pass straight through external static electric or magnetic fields without any deflection.

Ans: Accelerating (or oscillating) electric charges. A stationary charge creates only a static E-field; a constant velocity charge creates E and steady B fields; only an accelerating charge generates radiating EM waves.

Ans: Intensity $I = u \times c = (1.5 \times 10^{-8}) \times (3 \times 10^8) = 4.5\text{ W/m}^2$.

Ans: Radio waves $>$ Microwaves $>$ Ultraviolet (UV) rays $>$ X-rays.

Ans: Microwaves (specifically the UHF and SHF bands).

Ans: Because they are readily absorbed by water molecules in most materials, which massively increases their thermal motion (vibration), resulting in significant heating of the object.

Ans: Range: $\approx 400\text{ nm}$ to $700\text{ nm}$. Violet has the shortest wavelength and highest frequency, therefore Violet photons carry the highest energy ($E=hf$).

Ans: 1. Water purification (killing bacteria/germs in RO purifiers). 2. LASIK eye surgery (using extremely precise excimer UV lasers).

Ans: Produced by bombarding a heavy metal target (like Tungsten) with high-energy, rapidly decelerating electrons in a vacuum tube. Application: Medical radiography (imaging internal bone fractures).

Ans: Ultraviolet (UV) rays (specifically harmful UV-B and UV-C bands).

Ans: X-rays (Because their wavelength $\sim 1\text{ \AA}$ perfectly matches the interatomic spacing in crystals, producing clear diffraction patterns).

Ans: $f = \frac{c}{\lambda} = \frac{3 \times 10^8}{1.0 \times 10^{-10}} = 3 \times 10^{18}\text{ Hz}$.

Ans: The ionosphere reflects standard AM radio waves (sky waves) back toward the Earth, acting like a global mirror, which enables long-distance over-the-horizon radio communication.

Ans: Gamma ($\gamma$) rays.

Ans: Ultraviolet (UV) rays (Excimer Lasers).

Ans: Microwaves. Because of their shorter wavelength, they suffer far less diffraction (bending) around obstacles, allowing them to travel as tight, highly directional beams which are perfect for precise object location.

Ans: Welding arcs produce massive amounts of intensely bright visible light and highly dangerous **Ultraviolet (UV)** radiation. The special glass absorbs the UV rays, preventing permanent retinal damage (welder's flash).

Ans: $100\text{ MHz} = 10^8\text{ Hz}$. $\lambda = \frac{c}{f} = \frac{3 \times 10^8}{10^8} = 3\text{ meters}$.

Ans: No. Sound waves are mechanical, longitudinal compression waves that absolutely require a physical material medium (air, water, solid) to propagate. They are completely unrelated to oscillating electric and magnetic fields.