Q: A parallel plate capacitor is being charged by a current $$I. $$The current passes through a wire connected to one plate, but there is no conduction current between the plates. According to Maxwell's equations, what accounts for the continuity of current in the region between the plates?

The displacement current, defined as $$I_{\text{disp}} = \$$varepsilon_0$$ \dfrac{d\Phi_E}{dt}$$, ensures continuity.

Q: Which of the following is the correct mathematical form of the Ampère-Maxwell Law?

$$\oint \vec{B} \cdot d\vec{l} = \mu_0$ (I_{\text{through}} + \$$varepsilon_0$$ \dfrac{d\Phi_E}{dt})$$

Question 1

A parallel plate capacitor is being charged by a current $$I. $$The current passes through a wire connected to one plate, but there is no conduction current between the plates. According to Maxwell's equations, what accounts for the continuity of current in the region between the plates?

Accepted Answer

The displacement current, defined as $$I_{	ext{disp}} = \$$varepsilon_0$$ \dfrac{d\Phi_E}{dt}$$, ensures continuity.

Question 2

Which of the following is the correct mathematical form of the Ampère-Maxwell Law?

Accepted Answer

$$\oint \vec{B} \cdot d\vec{l} = \mu_0$ (I_{	ext{through}} + \$$varepsilon_0$$ \dfrac{d\Phi_E}{dt})$$

Question 3

A circular parallel plate capacitor with radius $$R is $$charging. The electric field between the plates is uniform and zero elsewhere. Three concentric circular surfaces A, B, and C have radii $$R/2$$, $$R$$, and $$3R/2$$ respectively. Rank the electric flux through these surfaces from least to greatest.

Accepted Answer

$$\Phi_C \u003c \Phi_A = \Phi_B$$

Question 4

A uniform magnetic field is directed out of the page and has a circular cross section of radius $$R. $$The magnitude of the magnetic field increases at a steady rate. What is the direction of the induced electric field at a point $$P on $$the edge of the region?

Accepted Answer

Tangential to the circle at $$P$$, in the direction that would produce a magnetic field into the page (clockwise).

Question 5

Consider a circular parallel plate capacitor with radius $$R. $$The magnetic field at a distance $$r$$ from the center between the plates is $$B(r). If$$ $$B_1 is $$the magnetic field at $$r = R$$, what is the magnetic field at $$r = R/2$$?

Accepted Answer

$$B_2$$ = \dfrac{1}{2} $$B_1$$

Question 6

Which of the following is NOT a correct statement about Maxwell's equations?

Accepted Answer

They require that the divergence of the electric field is always zero.

Question 7

A battery, two identical light bulbs, and two identical 9V batteries are connected as shown in a circuit diagram. Which of the following circuit diagrams correctly represents two bulbs in parallel, each powered by its own battery?

Accepted Answer

A diagram where each bulb is connected in parallel with a separate battery, and the bulbs are not in series with each other.

Question 8

According to Kirchhoff's junction rule, which of the following must be true at any junction in an electrical circuit?

Accepted Answer

The sum of currents entering the junction equals the sum of currents leaving the junction: $$\sum I_{	ext{in}} = \sum I_{	ext{out}}$$

Question 9

A current of $$2.0\,	ext{A}$$ flows in a circuit with two resistors ($$3\,\Omega$$ and $$1\,\Omega) $$and two batteries ($$+20\,	ext{V}$$ and $$+6\,	ext{V}) as $$shown. What is the emf $$\mathcal{E} of a $$third battery at the bottom of the circuit if the positive terminal is on the right?

Accepted Answer

$$+6\,	ext{V}$$

Question 10

Which of the following best describes the physical meaning of the displacement current in Maxwell's equations?

Accepted Answer

It accounts for the changing electric field in regions where there is no conduction current, such as between the plates of a charging capacitor.

Question 11

A generator is connected to a light bulb. Is it easier to turn the generator when the bulb is not connected, and why?

Accepted Answer

Yes, because when the bulb is not connected, no current flows and no electrical energy is converted to thermal energy.

Question 12

Which of the following is the correct integral form of Faraday's Law?

Accepted Answer

$$\oint \vec{E} \cdot d\vec{l} = -\dfrac{d\Phi_B}{dt}$$

Question 13

A microwave oven produces electromagnetic waves by which of the following mechanisms?

Accepted Answer

Oscillating electric currents in transmission antennas inside the oven.

Question 14

If observer A measures an electric field $$\vec{E}_A$$ and a magnetic field $$\vec{B}_A$$, and observer B moves at velocity $$\vec{v}_{BA}$$ relative to A, which of the following is the correct transformation for the electric field measured by B?

Accepted Answer

$$\vec{E}_B = \vec{E}_A + \vec{v}_{BA} 	imes \vec{B}_A$$

Question 15

Which of the following statements about the electric field induced by a changing magnetic field is correct?

Accepted Answer

The induced electric field forms closed loops and does not start or end on charges.

Question 16

A parallel plate capacitor is connected to a battery and is charging. At a given instant, the conduction current through the wires is $$I. $$What is the displacement current between the plates?

Accepted Answer

It is equal to $$I$$, so $$I_{	ext{disp}} = I.$$

Question 17

Which of the following best describes the role of the displacement current in the production of electromagnetic waves?

Accepted Answer

A time-varying electric field produces a displacement current, which in turn generates a magnetic field, allowing electromagnetic waves to propagate.

Question 18

Which of the following is a correct statement about the direction of the induced electric field when the magnetic flux through a loop increases out of the page?

Accepted Answer

The induced electric field circulates clockwise to oppose the increase in magnetic flux.

Electromagnetic Fields, Maxwell's Equations, and Fundamentals of Circuits ch 31

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