A coil is wrapped with 200 turns of wire
on the perimeter of a square frame of sides
20.5 cm. Each turn has the same area, equal
to that of the frame, and the total resistance
of the coil is 2.01 Ω. A uniform magnetic field
is turned on perpendicular to the plane of the
coil.
If the field changes linearly from 0 to
−0.0428 Wb/m2
in a time of 0.692 s, find
the magnitude of the induced emf in the coil
while the field is changing.
Answer in units of V.
To find the magnitude of the induced emf in the coil, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the induced emf in a coil is equal to the rate of change of magnetic flux through the coil.
First, we need to find the rate of change of magnetic flux. The magnetic flux can be calculated using the formula:
Φ = B * A
where Φ is the magnetic flux, B is the magnetic field, and A is the area of the coil.
Given that the magnetic field changes linearly from 0 to -0.0428 Wb/m² in a time of 0.692 s, we can calculate the rate of change of magnetic field (dB/dt) using the formula:
dB/dt = (B2 - B1) / (t2 - t1)
where B1 is the initial magnetic field, B2 is the final magnetic field, t1 is the initial time, and t2 is the final time.
Now, let's calculate the rate of change of magnetic field:
dB/dt = (-0.0428 Wb/m² - 0 Wb/m²) / (0.692 s - 0 s)
= -0.0428 Wb/m² / 0.692 s
≈ -0.0619 Wb/m²/s
Next, we need to calculate the area of the coil. Each turn of the coil has the same area as that of the square frame, which is given as 20.5 cm x 20.5 cm. However, the area needs to be converted to square meters:
A = (20.5 cm * 0.01 m/cm)^2
= 0.042025 m²
Now, let's calculate the rate of change of magnetic flux:
dΦ/dt = dB/dt * A
= (-0.0619 Wb/m²/s) * (0.042025 m²)
≈ -0.0026039 Wb/s
Finally, we can calculate the magnitude of the induced emf using Faraday's law:
ε = -dΦ/dt
Note that the negative sign indicates the direction of the induced current. Since the question asks for the magnitude of the induced emf, we can ignore the negative sign:
ε = |dΦ/dt|
= |-0.0026039 Wb/s|
≈ 0.0026039 V
Therefore, the magnitude of the induced emf in the coil, while the magnetic field is changing, is approximately 0.0026039 V.