Q: According to the Biot-Savart Law, what is the expression for the infinitesimal magnetic field $$d\vec{B} at $$point $$P$$ due to a current element $$i d\vec{s} at a $$distance $$\vec{r}$$ from $$P$$?

$$d\vec{B} = \frac{\mu_0$}{4\pi} \frac{i d\vec{s} \times \vec{r}}{r^3$}$$

Question 1

According to the Biot-Savart Law, what is the expression for the infinitesimal magnetic field $$d\vec{B} at $$point $$P$$ due to a current element $$i d\vec{s} at a $$distance $$\vec{r}$$ from $$P$$?

Accepted Answer

$$d\vec{B} = \frac{\mu_0$}{4\pi} \frac{i d\vec{s} 	imes \vec{r}}{r^3$}$$

Question 2

A long, straight wire carries a current $$i. $$Using Ampere’s Law, what is the magnitude of the magnetic field at a distance $$R$$ from the wire?

Accepted Answer

$$|\vec{B}| = \frac{\mu_0$ i}{2\pi R}$$

Question 3

In a coaxial cable with a solid inner conductor of radius $$a$$ and a conducting tube with inner radius $$b$$ and outer radius $$c$$, carrying equal and opposite currents $$I$$, what is the magnetic field at a distance $$r$$ such that $$a \u003c r \u003c b$$?

Accepted Answer

$$B = \frac{\mu_0$ I}{2\pi r}$$

Question 4

A battery is connected to a wire loop near a compass, as shown in the image. What is the physical principle demonstrated by the deflection of the compass needle?

Accepted Answer

A current-carrying wire produces a magnetic field that interacts with the compass needle.

Question 5

Which of the following best describes the direction of the magnetic force on a positive charge moving in a magnetic field?

Accepted Answer

It is perpendicular to both the velocity of the charge and the magnetic field, given by $$\vec{F} = q\vec{v} 	imes \vec{B}.$$

Question 6

According to Lenz’s Law, what is the direction of the induced current in a loop when the magnetic flux through the loop is increasing into the page?

Accepted Answer

The induced current will flow in a counterclockwise direction to oppose the increase.

Question 7

Which of the following is a correct statement of Faraday’s Law of Induction?

Accepted Answer

$$\mathcal{E} = -\frac{d\Phi_B}{dt}$$

Question 8

A solenoid has $$n$$ turns per unit length and carries a current $$i. $$What is the magnitude of the magnetic field inside the solenoid?

Accepted Answer

$$B = \mu_0$ n i$$

Question 9

Which of the following is NOT a way to change the magnetic flux through a loop and induce an emf?

Accepted Answer

Keeping the magnetic field and area constant.

Question 10

Maxwell’s correction to Ampere’s Law introduces the concept of displacement current. Which equation represents this correction?

Accepted Answer

$$\oint \vec{B} \cdot d\vec{s} = \mu_0$ i_{enc} + \mu_0$ \$$epsilon_0$$ \frac{d\Phi_E}{dt}$$

Question 11

Which of the following best describes the field lines of the induced electric field due to a changing magnetic field?

Accepted Answer

They form closed loops around the changing magnetic field.

Question 12

In an LC circuit, what is the angular frequency of oscillation?

Accepted Answer

$$\omega = \frac{1}{\sqrt{LC}}$$

Question 13

A resistor ($$R$$), inductor ($$L$$), and capacitor ($$C) $$are connected in series. Which of the following describes the behavior of the charge on the capacitor if the circuit is underdamped?

Accepted Answer

The charge oscillates with decreasing amplitude over time.

Question 14

Which of the following is true about electromagnetic waves as predicted by Maxwell’s equations?

Accepted Answer

They consist of oscillating electric and magnetic fields that propagate perpendicular to each other and to the direction of wave travel.

Question 15

A changing electric field between the plates of a capacitor produces which of the following?

Accepted Answer

A displacement current that generates a magnetic field.

Question 16

In a series RL circuit, the current increases according to $$I(t) = \frac{\mathcal{E}}{R} \left(1 - e$$^{-t/	au}$$ight). $$What is the time constant $$	au$$?

Accepted Answer

$$	au = \frac{L}{R}$$

Question 17

Which of the following statements about self-inductance is correct?

Accepted Answer

A changing current in a coil induces an emf that opposes the change in current.

Question 18

Which of the following best describes the energy transformation in an LC circuit?

Accepted Answer

Energy oscillates between the electric field of the capacitor and the magnetic field of the inductor.

Question 19

A square loop of wire is moving with velocity $$\vec{v}$$ into a region with a uniform magnetic field $$\vec{B}$$ directed into the page. What is the direction of the induced current in the loop as it enters the field?

Accepted Answer

Counterclockwise, to oppose the increase in magnetic flux.

Magnetic Fields, Induction, and Electromagnetic Oscillations

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