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?
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To determine the direction of the induced current, magnitude of the average induced emf, and the average induced current, we can use Faraday's law of electromagnetic induction.
1. Direction of the induced current:
According to Lenz's law, the direction of the induced current opposes the change that caused it. In this case, the loop's diameter is decreasing, which implies a decrease in the magnetic flux passing through the loop. To oppose this change, the induced current should create a magnetic field that opposes the external field, which is pointing into the paper. Therefore, the induced current will flow in a direction such that its magnetic field points out of the paper, counteracting the external field.
2. Magnitude of the average induced emf:
The average induced emf can be calculated using the formula:
emf = -N * ΔΦ / Δt
Where N is the number of turns in the loop, ΔΦ is the change in magnetic flux, and Δt is the change in time.
The change in magnetic flux can be determined by subtracting the initial flux from the final flux. Since the magnetic field is constant, the magnetic flux is given by:
Φ = B * A
Where B is the magnetic field and A is the area of the loop.
The initial area is calculated using the diameter:
A_initial = π * (D_initial / 2)^2
Substituting the values into the equation, we get:
A_initial = π * (0.178 m / 2)^2 = π * (0.089 m)^2
Similarly, the final area is calculated as:
A_final = π * (D_final / 2)^2 = π * (0.064 m / 2)^2 = π * (0.032 m)^2
Therefore, the change in magnetic flux is:
ΔΦ = B * (A_final - A_initial)
Substituting the values into the equation, we get:
ΔΦ = 0.78 T * (π * (0.032 m)^2 - π * (0.089 m)^2)
Finally, we can calculate the average induced emf:
emf = -N * ΔΦ / Δt
3. Average induced current:
The average induced current can be calculated using Ohm's law:
I = emf / R
Where emf is the average induced emf and R is the resistance of the coil.
Substituting the values into the equation, we can find the average induced current:
I = emf / 2.9 Ω
By following these steps, you can determine the direction of the induced current, the magnitude of the average induced emf, and the average induced current.