A 5.30 cm diameter wire coil is initially oriented so that its plane is perpendicular to a magnetic field of 0.570 T pointing up. During the course of 0.190 s, the field is changed to one of 0.350 T pointing down. What is the average induced emf in the coil?
Emf = RATE OF MAGNETIC FLUX CHANGE
= (Coil Area)*dB/dt
For dB/dt (the rate of B-field chnge, divide (0.57-(-0.19) T by 0.19 s
To calculate the average induced emf in the coil, we can use Faraday's law of electromagnetic induction, which states that the induced emf is equal to the rate of change of magnetic flux through the loop.
The formula for calculating the induced emf is:
emf = -d(Φ)/dt
Where:
emf is the induced electromotive force (emf)
d(Φ)/dt is the rate of change of magnetic flux
The magnetic flux (Φ) through the coil is given by the equation:
Φ = B * A * cos(θ)
Where:
B is the magnetic field strength (in T)
A is the area of the coil (in m²)
θ is the angle between the magnetic field and the normal to the coil's plane
Initially, when the coil is perpendicular to the magnetic field, the angle θ is 90 degrees. Therefore, cos(θ) = cos(90°) = 0.
With these equations, we can calculate the average induced emf:
1. Calculate the initial magnetic flux (Φ1) through the coil using the initial magnetic field strength and the coil's area:
Φ1 = B1 * A * cos(90°) = B1 * A * 0 = 0
2. Calculate the final magnetic flux (Φ2) through the coil using the final magnetic field strength and the coil's area:
Φ2 = B2 * A * cos(θ) = B2 * A * cos(0°) = B2 * A * 1 = B2 * A
3. Calculate the rate of change of magnetic flux (dΦ/dt) by subtracting the initial magnetic flux from the final magnetic flux and dividing it by the time interval:
dΦ/dt = (Φ2 - Φ1) / Δt = (B2 * A - 0) / Δt = B2 * A / Δt
4. Calculate the average induced emf (emf) by multiplying the rate of change of magnetic flux by -1:
emf = -dΦ/dt = -(B2 * A / Δt)
Now we just need to substitute the given values:
- B2 = 0.350 T (final magnetic field strength)
- A = π * (d/2)² = π * (5.30 cm/2)² (area of the coil)
- Δt = 0.190 s (time interval)
- Convert the diameter from centimeters to meters:
d = 5.30 cm = 5.30 cm * (1 m/100 cm) = 0.053 m
- Calculate the area of the coil:
A = π * (0.053 m/2)² = π * 0.02809 m²
Now, we can substitute these values into the equation to find the average induced emf:
emf = -(B2 * A / Δt) = -(0.350 T * π * 0.02809 m² / 0.190 s)
Calculating this expression will give you the average induced emf in the coil.