Question 1

Which of the following statements best distinguishes between electrostatic and induced electric fields?

Accepted Answer

Electrostatic fields are conservative and derived from electric potential, while induced electric fields are non-conservative and not derived from potential.

Question 2

According to Faraday’s law, what is the induced emf around a closed loop enclosing a time-varying magnetic flux $$\Phi_m$$?

Accepted Answer

$$\displaystyle \mathcal{E} = -\frac{d\Phi_m}{dt}$$

Question 3

A uniform magnetic field inside a solenoid increases at a constant rate. What is the expression for the magnitude of the induced electric field at a distance $$r$$ from the axis, inside the solenoid?

Accepted Answer

$$\displaystyle E = \frac{r}{2} \left| \frac{dB}{dt} \right|$$

Question 4

Which of the following is the correct form of Maxwell’s generalization of Ampère’s law in vacuum?

Accepted Answer

$$\displaystyle \oint d\vec{s} \cdot \vec{B} = \mu_0$ \left( I + \$$epsilon_0$$ \frac{d\Phi_e}{dt} \right)$$

Question 5

A parallel-plate capacitor is being charged. Which term in Maxwell’s equations accounts for the magnetic field between the plates?

Accepted Answer

The displacement current term $$\displaystyle \$$epsilon_0$$ \frac{d\Phi_e}{dt}$$

Question 6

A circular loop of radius $$r is $$placed in a region where the magnetic field increases uniformly with time. What is the direction of the induced electric field at a point on the loop?

Accepted Answer

Tangential to the loop, forming closed circles around the axis

Question 7

In the context of the displacement current, why must Ampère’s law be modified for a charging capacitor?

Accepted Answer

Because the conduction current is zero between the plates, but the changing electric field still produces a magnetic field

Question 8

A long solenoid has $$n = 400$$ turns/m and radius $$R = 1.1 cm. $$The current increases at a rate $$\frac{di}{dt} = k. At a $$point outside the solenoid, at distance $$3.50 cm $$from its axis, the induced electric field is $$8 	imes 10$$^{-6}$$ V/m. What is the value of k$?

Accepted Answer

$$k = 0.089 A/s$$

Question 9

Which of the following best describes the solution to the paradox regarding the induced electric field at the intersection of two loops in an infinite uniform magnetic field?

Accepted Answer

The paradox is resolved by considering that real magnetic fields are generated by finite-sized currents, and the induced electric field must respect the symmetry and boundary conditions of the physical system.

Question 10

A straight wire carries a current $$I(t) = kt. $$Using the quasistatic approximation, what is the induced electric field at a distance $$x$$ from the wire?

Accepted Answer

$$\displaystyle E_{ind}(x) = -\frac{\mu_0$ k}{2\pi} \ln \frac{x}{a}$$

Question 11

Which of the following is NOT a Maxwell equation in vacuum?

Accepted Answer

$$\displaystyle \oint d\vec{s} \cdot \vec{E} = \mu_0$ I$$

Question 12

A rectangular loop of wire with mass $$m$$ and resistance $$R is $$pulled out of a region with a uniform magnetic field $$B by a $$constant force $$F_{ext}. $$Which of the following best describes the loop’s motion after 1 second?

Accepted Answer

The loop may leave the magnetic field region depending on the balance between the applied force and the magnetic damping force due to induced current.

Question 13

Which of the following is a consequence of neglecting the finite speed of electromagnetic propagation (the quasistatic approximation) in the calculation of induced electric fields?

Accepted Answer

The induced electric field can become infinite at large distances.

Question 14

A parallel-plate capacitor is kept at a fixed voltage $$V$$ while the plates are moved apart with velocity $$u. $$What is the expression for the magnetic field inside the capacitor?

Accepted Answer

$$\displaystyle B = \frac{\mu_0$ \$$epsilon_0$$ V u}{2 \pi r d^2$}$$

Question 15

Which of the following best describes the field lines of an induced electric field generated by a time-varying magnetic field?

Accepted Answer

They form closed loops.

Question 16

Which of the following is the correct expression for the displacement current through a surface $$S$$?

Accepted Answer

$$\displaystyle I_d = \$$epsilon_0$$ \frac{d}{dt} \int_S d\vec{A} \cdot \vec{E}$$

Induced Electric and Magnetic Fields, Maxwell’s Equations, and Paradoxes

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