Maxwell Equations and Potentials - Electromagnetic theory - Physics Previous Year Questions

1

2007 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2007

A charged capacitor (C) is connected in series with an inductor (L). When the displacement current reduces to zero, the energy of the LC circuit is

A

stored entirely in its magnetic field.

B

stored entirely in its electric field.

C

distributed equally among its electric and magnetic fields.

D

radiated out of the circuit.

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2

2007 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2007

The electromagnetic field due to a point charge must be described by Liénard-Weichert potentials when

A

the point charge is highly accelerated.

B

the electric and magnetic fields are not perpendicular.

C

the point charge is moving with velocity close to that of light.

D

the calculation is done for the radiation zone, i.e. far away from the charge.

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3

2007 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2007

A vector field is defined everywhere as \(\vec{F} = \frac{y^3}{L} \hat{i} + z \hat{k}\). The net flux of \(\vec{F}\) associated with a cube of side \(L\), with one vertex at the origin and sides along the positive X, Y, and Z axes, is

A

\(2L^3\)

B

\(4L^3\)

C

\(8L^3\)

D

\(10L^3\)

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4

2007 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2007

A toroidal coil has \( N \) closely-wound turns. Assume the current through the coil to be \( I \) and the toroid is filled with a magnetic material of relative permittivity \( \mu_r \). The magnitude of magnetic induction \( \vec{B} \) inside the toroid, at a radial distance \( r \) from the axis, is given by

A

\(\mu_r \mu_0 N I r \)

B

\(\frac{\mu_r \mu_0 N I}{r} \)

C

\(\frac{\mu_r \mu_0 N I}{2 \pi r} \)

D

\(2 \pi \mu_r \mu_0 N I r \)

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5

2007 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2007

Can the following scalar and vector potentials describe an electromagnetic field? \( \phi(\vec{x}, t) = 3xyz - 4t \) \( \vec{A}(\vec{x}, t) = (2x - \omega t)\hat{i} + (y - 2z)\hat{j} + (z - 2xe^{-\omega t})\hat{k} \) where \( \omega \) is a constant.

A

Yes, in the Coulomb gauge.

B

Yes, in the Lorentz gauge.

C

Yes, provided \(\omega = 0 \).

D

No.

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6

2007 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2007

Can the following scalar and vector potentials describe an electromagnetic field? \( \phi(\vec{x}, t) = 3xyz - 4t \) \( \vec{A}(\vec{x}, t) = (2x - \omega t) \hat{i} + (y - 2z) \hat{j} + (z - 2x e^{-i \omega t}) \hat{k} \) where \( \omega \) is a constant.

A

Yes, in the Coulomb gauge.

B

Yes, in the Lorentz gauge.

C

Yes, provided \(\omega = 0 \).

D

No.

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7

2008 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2008

The curl of a vector field \(\vec{F}\) is \(2\hat{x}\). Identify the appropriate vector field \(\vec{F}\) from the choices given below.

A

\(\vec{F} = 2z\hat{x} + 3z\hat{y} + 5y\hat{z}\)

B

\(\vec{F} = 3z\hat{y} + 5y\hat{z}\)

C

\(\vec{F} = 3y\hat{y} + 5y\hat{z}\)

D

\(\vec{F} = 2\hat{x} + 5y\hat{z}\)

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8

2008 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2008

A rod of length \( L \) with uniform charge density \( \lambda \) per unit length is in the xy-plane and rotating about z-axis passing through one of its edge with an angular velocity \( \vec{\omega} \) as shown in the figure below. \( \{\hat{r}, \hat{\phi}, \hat{z}\} \) refer to the unit vectors at Q, \( \vec{A} \) is the vector potential at a distance \( d \) from the origin O along z-axis for \( d >> L \) and \( \vec{J} \) is the current density due to the motion of the rod. Which one of the following statements is correct?

A

\(\vec{J} \) along \(\hat{r} \); \(\vec{A} \) along \(\hat{z} \); \(|\vec{A}| \propto \frac{1}{d} \)

B

\(\vec{J} \) along \(\hat{\phi} \); \(\vec{A} \) along \(\hat{\phi} \); \(|\vec{A}| \propto \frac{1}{d^2} \)

C

\(\vec{J} \) along \(\hat{r} \); \(\vec{A} \) along \(\hat{z} \); \(|\vec{A}| \propto \frac{1}{d^2} \)

D

\(\vec{J} \) along \(\hat{\phi} \); \(\vec{A} \) along \(\hat{\phi} \); \(|\vec{A}| \propto \frac{1}{d} \)

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9

2008 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2008

At time \(t = 0\), a charge distribution \(\rho(\vec{r}, 0)\) exists within an ideal homogeneous conductor of permittivity \(\epsilon\) and conductivity \(\sigma\). At a later time \(\rho(\vec{r}, t)\) is given by

A

\(\rho(\vec{r}, t) = \rho(\vec{r}, 0) \exp\left(-\frac{\sigma t}{\epsilon}\right)\)

B

\(\rho(\vec{r}, t) = \frac{\rho(\vec{r}, 0)}{1 + (\sigma t / \epsilon)^2}\)

C

\(\rho(\vec{r}, t) = \rho(\vec{r}, 0) \exp\left[-\left(\frac{\sigma t}{\epsilon}\right)^2\right]\)

D

\(\rho(\vec{r}, t) = \rho(\vec{r}, 0) \frac{\epsilon}{\sigma t} \sin\left(\frac{\sigma t}{\epsilon}\right)\)

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10

2009 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2009

A conducting loop L of surface area S is moving with a velocity \(\vec{v}\) in a magnetic field \(\vec{B}(\vec{r}, t) = B_0 \hat{y}\), \(B_0\) is a positive constant of suitable dimensions. The emf induced, \(V_{emf}\), in the loop is given by

A

\(- \int_S \frac{\partial \vec{B}}{\partial t} \cdot d\vec{S}\)

B

\(\oint_L (\vec{v} \times \vec{B}) \cdot d\vec{L}\)

C

\(- \int_S \frac{\partial \vec{B}}{\partial t} \cdot d\vec{S} - \oint_L (\vec{v} \times \vec{B}) \cdot d\vec{L}\)

D

\(- \int_S \frac{\partial \vec{B}}{\partial t} \cdot d\vec{S} + \oint_L (\vec{v} \times \vec{B}) \cdot d\vec{L}\)

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11

2009 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2009

The magnetic field (in A m⁻¹) inside a long solid cylindrical conductor of radius a = 0.1 m is, \(\vec{H} = \frac{10^3}{r}\left[\frac{1}{\alpha^2}\sin(\alpha r) - \frac{r}{\alpha}\cos(\alpha r)\right]\hat{\phi}\), where \(\alpha = \frac{\pi}{2a}\). What is the total current (in A) in the conductor?

A

\(\frac{\pi}{2a}\)

B

\(\frac{800}{\pi}\)

C

\(\frac{400}{\pi}\)

D

\(\frac{300}{\pi}\)

Check answer

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12

2009 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2009

Which one of the following current densities, \(\vec{J}\), can generate the magnetic vector potential \(\vec{A} = (y^2\hat{i} + x^2\hat{j})\)?

A

\(\frac{2}{\mu_0}(x\hat{i} + y\hat{j})\)

B

\(-\frac{2}{\mu_0}(\hat{i} + \hat{j})\)

C

\(\frac{2}{\mu_0}(\hat{i} - \hat{j})\)

D

\(\frac{2}{\mu_0}(x\hat{i} - y\hat{j})\)

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13

2009 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2009

In a non-conducting medium characterized by \(\varepsilon = \varepsilon_0\), \(\mu = \mu_0\) and conductivity \(\sigma = 0\), the electric field (in V m⁻¹) is given by \(\vec{E} = 20 \sin[10^8 t - kz]\hat{j}\). The magnetic field, \(\vec{H}\) (in A m⁻¹), is given by

A

\(20k \cos[10^8 t - kz]\hat{i}\)

B

\(\frac{20k}{10^8 \mu_0} \sin[10^8 t - kz]\hat{j}\)

C

\(-\frac{20k}{10^8 \mu_0} \sin[10^8 t - kz]\hat{i}\)

D

\(-20k \cos[10^8 t - kz]\hat{j}\)

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14

2010 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2010

Consider the operations \( P: \vec{r} \rightarrow -\vec{r} \) (parity) and \( T: t \rightarrow -t \) (time-reversal). For the electric and magnetic fields \( \vec{E} \) and \( \vec{B} \), which of the following set of transformations is correct?

A

\(P: \vec{E} \rightarrow -\vec{E}, \vec{B} \rightarrow \vec{B}; \; T: \vec{E} \rightarrow \vec{E}, \vec{B} \rightarrow -\vec{B} \)

B

\(P: \vec{E} \rightarrow \vec{E}, \vec{B} \rightarrow \vec{B}; \; T: \vec{E} \rightarrow \vec{E}, \vec{B} \rightarrow \vec{B} \)

C

\(P: \vec{E} \rightarrow -\vec{E}, \vec{B} \rightarrow \vec{B}; \; T: \vec{E} \rightarrow -\vec{E}, \vec{B} \rightarrow -\vec{B} \)

D

\(P: \vec{E} \rightarrow \vec{E}, \vec{B} \rightarrow -\vec{B}; \; T: \vec{E} \rightarrow -\vec{E}, \vec{B} \rightarrow \vec{B} \)

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15

2010 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2010

Consider a conducting loop of radius a and total loop resistance R placed in a region with a magnetic field B thereby enclosing a flux φ₀. The loop is connected to an electronic circuit as shown, the capacitor being initially uncharged.
If the loop is pulled out of the region of the magnetic field at a constant speed u, the final output voltage V_out is independent of

A

φ₀

B

u

C

R

D

C

Check answer

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16

2010 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2010

For a plane wave of angular frequency \(\omega\) and propagation vector \(\vec{k}\) propagating in the medium Maxwell's equations reduce to

A

\(\vec{k} \cdot \vec{E} = 0; \quad \vec{k} \cdot \vec{H} = 0; \quad \vec{k} \times \vec{E} = \omega \varepsilon \vec{H}; \quad \vec{k} \times \vec{H} = -\omega \mu \vec{E}\)

B

\(\vec{k} \cdot \vec{E} = 0; \quad \vec{k} \cdot \vec{H} = 0; \quad \vec{k} \times \vec{E} = -\omega \varepsilon \vec{H}; \quad \vec{k} \times \vec{H} = \omega \mu \vec{E}\)

C

\(\vec{k} \cdot \vec{E} = 0; \quad \vec{k} \cdot \vec{H} = 0; \quad \vec{k} \times \vec{E} = -\omega \mu \vec{H}; \quad \vec{k} \times \vec{H} = \omega \varepsilon \vec{E}\)

D

\(\vec{k} \cdot \vec{E} = 0; \quad \vec{k} \cdot \vec{H} = 0; \quad \vec{k} \times \vec{E} = \omega \mu \vec{H}; \quad \vec{k} \times \vec{H} = -\omega \varepsilon \vec{E}\)

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17

2011 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2011

A uniform surface current is flowing in the positive \(y\)-direction over an infinite sheet lying in \(x-y\) plane. The direction of the magnetic field is

A

along \(\hat{i}\) for \(z > 0\) and along \(-\hat{i}\) for \(z < 0\)

B

along \(\hat{k}\) for \(z > 0\) and along \(-\hat{k}\) for \(z < 0\)

C

along \(-\hat{i}\) for \(z > 0\) and along \(\hat{i}\) for \(z < 0\)

D

along \(-\hat{k}\) for \(z > 0\) and along \(\hat{k}\) for \(z < 0\)

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18

2011 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2011

Which of the following expressions for a vector potential \(\vec{A}\) DOES NOT represent a uniform magnetic field of magnitude \(B_0\) along the \(z\)-direction?

A

\(\vec{A} = (0, B_0 x, 0)\)

B

\(\vec{A} = (-B_0 y, 0, 0)\)

C

\(\vec{A} = \left(\frac{B_0 x}{2}, \frac{B_0 y}{2}, 0\right)\)

D

\(\vec{A} = \left(-\frac{B_0 y}{2}, \frac{B_0 x}{2}, 0\right)\)

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19

2011 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2011

A uniform surface current is flowing in the positive y-direction over an infinite sheet lying in x-y plane. The direction of the magnetic field is

A

along \(\hat{i}\) for \(z > 0\) and along \(-\hat{i}\) for \(z < 0\)

B

along \(\hat{k}\) for \(z > 0\) and along \(-\hat{k}\) for \(z < 0\)

C

along \(-\hat{i}\) for \(z > 0\) and along \(\hat{i}\) for \(z < 0\)

D

along \(-\hat{k}\) for \(z > 0\) and along \(\hat{k}\) for \(z < 0\)

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20

2011 · Physics · Electromagnetic theory · Maxwell Equations and Potentials

Physics (PH) 2011

Which of the following expressions for a vector potential \(\vec{A}\) DOES NOT represent a uniform magnetic field of magnitude \(B_0\) along the z-direction?

A

\(\vec{A} = (0, B_0 x, 0)\)

B

\(\vec{A} = (-B_0 y, 0, 0)\)

C

\(\vec{A} = \left(\frac{B_0 x}{2}, \frac{B_0 y}{2}, 0\right)\)

D

\(\vec{A} = \left(-\frac{B_0 y}{2}, \frac{B_0 x}{2}, 0\right)\)

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