A circular loop in the plane of the paper lies in a 0.78 T magnetic field pointing into the paper.

If the loop's diameter changes from 17.8 cm to 6.4 cm in 0.22 s, what is the direction of the induced current?
What is the magnitude of the average induced emf?
If the coil resistance is 2.9 Ω, what is the average induced current?

The induced current will be in the dorection that increases the flux through the loop.

The magnitude of the induced emf is
d(Phi)/dt = B*dA/dt
= 0.78T*(pi/4)[(.178^2 -.064)^2]/0.22s
= 0.077 V

Divide by 2.9 ohms for the induced current.

Thank you!

To find the direction of the induced current, we can use the right-hand rule for electromagnetic induction. According to the right-hand rule, if you point your right thumb in the direction of the magnetic field (into the paper in this case), and curl your fingers in the direction of the changing magnetic field (from the larger diameter to the smaller diameter), then your fingers will point in the direction of the induced current.

So, using the right-hand rule, the direction of the induced current in the loop is counterclockwise.

To find the magnitude of the average induced emf, we can use Faraday's Law of electromagnetic induction. Faraday's Law states that the induced emf (ε) is equal to the negative rate of change of magnetic flux (ϕ) through the loop. Mathematically, it can be expressed as:

ε = -dϕ/dt

The magnetic flux through the loop can be calculated as the product of the magnetic field (B) and the area (A) of the loop. In this case, the loop is circular, so the area can be calculated using the formula for the area of a circle (A = πr^2), where r is the radius of the loop (half the diameter).

Initially, the radius (r1) of the loop is half of the initial diameter (17.8 cm / 2 = 8.9 cm = 0.089 m), and the area (A1) of the loop is π(0.089 m)^2.

Finally, the final radius (r2) of the loop is half of the final diameter (6.4 cm / 2 = 3.2 cm = 0.032 m), and the area (A2) of the loop is π(0.032 m)^2.

Plugging these values into the formula for the magnetic flux (ϕ = BA), we can find the rate of change of magnetic flux (dϕ/dt), which will give us the average induced emf (ε).

Using the given time interval (0.22 s), the average induced emf can be calculated.

To find the average induced current, we can use Ohm's Law (V = IR), where V is the induced emf (ε) and R is the resistance of the coil.

Plugging in the values for the average induced emf (ε) and the coil resistance (R), we can calculate the average induced current (I).

Note: It is assumed that the resistance of the coil remains constant and the changing diameter does not affect it significantly.

Let's calculate the magnitude of the average induced emf and the average induced current step by step.

To determine the direction of the induced current, we can use Lenz's Law. Lenz's Law states that the direction of the induced current in a closed loop is such that it opposes the change in the magnetic field that produces it.

Since the magnetic field is pointing into the paper, a decrease in the loop's diameter would result in an increase in the magnetic flux through the loop. To oppose this increase, the induced current would create a magnetic field that points out of the paper. Therefore, the direction of the induced current would be counterclockwise when viewed from above.

To find the magnitude of the average induced emf, we can use Faraday's Law of electromagnetic induction. Faraday's Law states that the induced emf is equal to the rate of change of magnetic flux through the loop.

The magnetic flux (Φ) is given by the product of the magnetic field (B) and the area (A) of the loop: Φ = B * A.

The rate of change of the magnetic flux is given by ΔΦ/Δt, where ΔΦ is the change in magnetic flux and Δt is the time interval.

The change in magnetic flux is the difference between the initial and final magnetic flux: ΔΦ = Φ_final - Φ_initial.

The initial magnetic flux can be calculated using the initial diameter (d1) of the loop: Φ_initial = B * π * (d1/2)^2.

The final magnetic flux can be calculated using the final diameter (d2) of the loop: Φ_final = B * π * (d2/2)^2.

Substituting these values, we can find the change in magnetic flux.

Once we have the change in magnetic flux and the time interval, we can calculate the average induced emf using the formula: ε = ΔΦ/Δt.

Finally, to find the average induced current, we can use Ohm's Law: I = ε/R, where R is the resistance of the loop.

By substituting the values into the formulas, we can find the answers to the remaining questions.