Physics with Calculus: Step-by-Step Guidance for PHYS 1402 Exam 2 Practice Questions

Q1. A wire loop is turned so its surface faces a uniform magnetic field more directly. What happens to the magnitude of the magnetic flux?

Background

Topic: Magnetic Flux and Induction

This question tests your understanding of how the orientation of a loop relative to a magnetic field affects the magnetic flux through the loop.

Key Terms and Formulas

• Magnetic flux ($\Phi_B$): The measure of the magnetic field passing through a given area.

• Formula: $\Phi_B = B A \cos(\theta)$

• $B$ = magnetic field strength

• $A$ = area of the loop

• $\theta$ = angle between the magnetic field and the normal (perpendicular) to the loop's surface

Step-by-Step Guidance

1. Identify the initial orientation: The loop is initially at some angle $\theta$ with respect to the magnetic field.

2. As the loop turns to face the field more directly, $\theta$ decreases (approaches $0^\circ$).

3. Recall that $\cos(\theta)$ increases as $\theta$ decreases from $90^\circ$ to $0^\circ$.

4. Since $B$ and $A$ are constant, the only changing factor in $\Phi_B = B A \cos(\theta)$ is $\cos(\theta)$.

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Final Answer: (A) Increases

As the loop faces the field more directly, $\theta$ decreases, so $\cos(\theta)$ increases, and thus the magnetic flux increases.

Q2. A closed conducting ring sits in a steady, uniform magnetic field without moving or changing shape. Is an induced current sustained by this field alone?

Background

Topic: Faraday's Law of Induction

This question tests your understanding of the conditions required to induce a current in a closed loop.

Key Terms and Formulas

• Faraday's Law: $\mathcal{E} = -N \frac{d\Phi_B}{dt}$

• Induced emf ($\mathcal{E}$) depends on the rate of change of magnetic flux through the loop.

• For a current to be induced, $\frac{d\Phi_B}{dt}$ must not be zero.

Step-by-Step Guidance

1. Check if the magnetic field ($B$) is changing with time. In this case, it is steady (not changing).

2. Check if the area ($A$) or orientation ($\theta$) of the loop is changing. Here, both are fixed.

3. Since $\Phi_B = B A \cos(\theta)$ and none of $B$, $A$, or $\theta$ are changing, $\frac{d\Phi_B}{dt} = 0$.

4. Recall that a nonzero induced current requires a nonzero induced emf.

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Final Answer: (C) No

With no change in magnetic flux, there is no induced emf and thus no sustained induced current.

Q3. A conducting rectangular loop expands while a field pointing into the page stays uniform. Viewed from the front, which induced current direction opposes the change?

Background

Topic: Lenz's Law and Faraday's Law

This question tests your ability to determine the direction of induced current that opposes a change in magnetic flux.

Key Terms and Formulas

• Lenz's Law: The induced current creates a magnetic field that opposes the change in flux.

• Faraday's Law: $\mathcal{E} = -N \frac{d\Phi_B}{dt}$

• Right-hand rule: Used to determine the direction of the induced current and its associated magnetic field.

Step-by-Step Guidance

1. As the loop expands, its area $A$ increases, so the magnetic flux $\Phi_B = B A$ increases (since $B$ is into the page and constant).

2. The increase in into-the-page flux means the induced current will try to oppose this increase.

3. To oppose an increasing into-the-page flux, the induced current must create an out-of-the-page magnetic field.

4. Use the right-hand rule: Curl your fingers in the direction of the current; your thumb points in the direction of the induced field.

Try solving on your own before revealing the answer!

Final Answer: (B) Counterclockwise

A counterclockwise current (viewed from the front) produces an out-of-the-page field, opposing the increase in into-the-page flux.

Q4. A loop is drawn out of a region where the magnetic field points into the page. Its field-threaded area shrinks. Which induced field does the loop try to create?

Background

Topic: Lenz's Law

This question tests your understanding of how the induced field direction is determined when the magnetic flux through a loop decreases.

Key Terms and Formulas

• Lenz's Law: The induced current creates a magnetic field that opposes the change in flux.

• Magnetic flux: $\Phi_B = B A$ (since $\theta = 0$ for perpendicular field)

Step-by-Step Guidance

1. As the loop is pulled out, the area $A$ exposed to the into-the-page field decreases, so the into-the-page flux decreases.

2. The induced current will try to oppose this decrease in into-the-page flux.

3. To oppose a decrease, the induced field must point in the same direction as the original field (into the page).

4. Think about the direction of current needed to produce this induced field using the right-hand rule.

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Final Answer: (B) Into page

The induced field points into the page to oppose the decrease in into-the-page flux as the loop leaves the field region.

Q5. Two metal rings with equal area experience the same changing magnetic flux, but one ring has greater resistance. Which comparison is correct?

Background

Topic: Faraday's Law and Ohm's Law

This question tests your understanding of how resistance affects induced emf and current in loops experiencing the same changing flux.

Key Terms and Formulas

• Faraday's Law: $\mathcal{E} = -N \frac{d\Phi_B}{dt}$

• Ohm's Law: $I = \frac{\mathcal{E}}{R}$

• Emf ($\mathcal{E}$) depends only on the rate of change of flux and number of turns, not resistance.

Step-by-Step Guidance

1. Both rings have the same area and experience the same $\frac{d\Phi_B}{dt}$, so both have the same induced emf.

2. The ring with greater resistance will have a smaller current, since $I = \frac{\mathcal{E}}{R}$.

3. Compare the emf and current in both rings based on these relationships.

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Final Answer: (B) Emfs are equal but currents differ

Both rings have the same induced emf, but the ring with greater resistance carries less current.

Q6. A student doubles the rate at which flux changes through a stationary coil without changing its turns. What happens to the magnitude of its induced emf?

Background

Topic: Faraday's Law

This question tests your understanding of how the rate of change of magnetic flux affects the induced emf in a coil.

Key Terms and Formulas

• Faraday's Law: $\mathcal{E} = -N \frac{d\Phi_B}{dt}$

• Magnitude of emf is proportional to the rate of change of flux.

Step-by-Step Guidance

1. Let the original rate of change of flux be $\frac{d\Phi_B}{dt}$, and the original emf be $\mathcal{E}_0 = N \left| \frac{d\Phi_B}{dt} \right|$.

2. If the rate is doubled, the new rate is $2 \frac{d\Phi_B}{dt}$.

3. Substitute this new rate into Faraday's Law to find the new emf.

4. Compare the new emf to the original emf to see how it changes.

Try solving on your own before revealing the answer!

Final Answer: (B) Doubles

The induced emf is directly proportional to the rate of change of flux, so doubling the rate doubles the emf.