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Answer :
Faraday's Law (ΔΦ = -εΔt) & avg change in flux give |ε_avg| = (ΔB * πr²) / Δt ≈ 0.157 V due to perpendicular B field change.
1. Faraday's Law of Electromagnetic Induction:
Faraday's Law states that the induced electromotive force (EMF) in a closed loop is equal to the negative rate of change of the magnetic flux (Φ) through the loop:
EMF = -ΔΦ / Δt
where:
EMF is the electromotive force in volts (V)
ΔΦ is the change in magnetic flux through the loop in webers (Wb)
Δt is the time interval in seconds
2. Magnetic Flux:
The magnetic flux through the coil depends on the magnetic field strength (B), the area (A) of the coil perpendicular to the field, and the cosine of the angle (θ) between the field and the normal to the area:
Φ = B * A * cos(θ)
In this scenario, the magnetic field is perpendicular to the plane of the coil (θ = 0°), so cos(θ) = 1.
3. Area of the Coil:
The area of the coil can be calculated using the formula for the area of a circle:
A = π * r²
where r is the radius of the coil.
4. Change in Magnetic Field:
We are given the initial magnetic field strength (B_initial) = 52.3 mT = 0.0523 T, the final magnetic field strength (B_final) = 97.7 mT = 0.0977 T, and the time interval (Δt) = 0.125 s.
The change in magnetic field (ΔB) is:
ΔB = B_final - B_initial = 0.0977 T - 0.0523 T = 0.0454 T
5. Average EMF:
Since we are looking for the average EMF during the time interval, we can assume the change in magnetic flux is constant over that period. Therefore, the average EMF (ε_avg) is:
ε_avg = - ΔΦ / Δt
Substitute the expressions for ΔΦ and Δt:
ε_avg = - (B_final - B_initial) * A * cos(θ) / Δt
Plug in the known values:
ε_avg = - (0.0454 T) * π * (2.71 cm x 10⻲ m)² * cos(0°) / (0.125 s)
Note: Convert centimeters to meters for consistency in units.
Calculate the magnitude (ignoring the negative sign):
|ε_avg| ≈ 0.157 V
Therefore, the magnitude of the average EMF induced in the coil is approximately 0.157 volts.
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