AP Physics 2 Exam Question:

A circular loop with radius R is lying in the xy-plane. A uniform magnetic field B is directed along the positive z-axis. The loop is initially in the plane x = 0 and carries a current I in the counterclockwise direction as seen from the positive z-axis. The loop is pulled to the right with a constant velocity v, perpendicular to the plane of the loop, by an external force. Determine the induced electric field and the induced current in the loop.

Solution:

To find the induced electric field and current in the loop, we can use Faraday's Law and Lenz's Law.

Step 1: Determining the magnetic flux:

The magnetic flux (Φ) through the loop is given by:

Φ = B * A * cosθ

where B is the magnetic field, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the loop.

In this case, the loop lies in the xy-plane, so the normal to the loop is along the positive z-axis and is perpendicular to the magnetic field. Therefore, θ = 0, and the cosθ term becomes 1.

The area of the loop is A = π * R^2.

So, the magnetic flux through the loop is:

Φ = B * π * R^2

Step 2: Applying Faraday's Law:

Faraday's Law states that the electric field induced around a closed loop is equal to the rate of change of magnetic flux through the loop.

Therefore, the induced electric field (E) is given by:

E = -(dΦ/dt)

where dΦ/dt is the rate of change of magnetic flux.

Differentiating Φ with respect to time, we get:

(dΦ/dt) = (dB/dt) * π * R^2

Step 3: Determining the direction of the induced electric field:

According to Lenz's Law, the induced current will create a magnetic field to oppose the change in the applied magnetic field.

Since the loop is being pulled to the right, the applied magnetic field will increase in the downward direction through the loop. Therefore, the induced current will create a magnetic field pointing upward to oppose this change.

By applying the right-hand rule, we can determine that the induced current I in the loop will flow counterclockwise.

Step 4: Finding the induced current:

To find the induced current in the loop, we use Ohm's Law:

E = I * R

where E is the induced electric field and R is the resistance of the loop.

Substituting the value of E we found in Step 2, we have:

-(dΦ/dt) = I * R

Simplifying, we get:

-(dB/dt) * π * R^2 = I * R

R cancels out, and we can rearrange the equation to solve for I:

I = -(dB/dt) * π * R

Answer:

Therefore, the induced electric field in the loop is -(dB/dt) * π * R^2, and the induced current in the loop is -(dB/dt) * π * R (counterclockwise direction).