A circular coil (660 turns, radius = 0.037 m) is rotating in a uniform magnetic field. At t = 0 s, the normal to the coil is perpendicular to the magnetic field. At t = 0.024 s, the normal makes an angle of 45o with the field because the coil has made one-eighth of a revolution. An average emf of magnitude 0.095 V is induced in the coil. Find the magnitude of the magnetic field at the location of the coil.
emf = N d flux /dt
emf = 660 B pi r^2 d/dt (sin 2 pi t/T)
when t = .024, t/T = 1/8
T = .192 second
so
emf = -660 (pi r^2) (2 pi/T)B cos (2 pi t/T)
now I do not know what "average means. Obviously the average of a cosine function is zero.
You might guess they mean magnitude in which case
.095 = 660(pi r^2)(2 pi/T) B
or they might mean root mean square or something in which case multiply that B by 1.414
To find the magnitude of the magnetic field at the location of the coil, we can use Faraday's Law of electromagnetic induction. This law states that the magnitude of the induced electromotive force (emf) is equal to the rate of change of magnetic flux through a closed loop.
The formula for Faraday's Law is:
emf = -N(dΦ/dt)
Where:
emf is the induced electromotive force (V)
N is the number of turns in the coil (660 turns)
dΦ/dt is the rate of change of magnetic flux through the coil (T·m²/s)
In this case, the coil is rotating and we want to find the magnitude of the magnetic field at the location of the coil. We know the following information:
- Number of turns in the coil (N) = 660 turns
- Radius of the coil (r) = 0.037 m
- Time interval (Δt) = 0.024 s
- Angle between the normal to the coil and the magnetic field (θ) = 45°
First, let's find the change in magnetic flux (ΔΦ):
ΔΦ = B * A * cos(θ)
Where:
B is the magnitude of the magnetic field (T)
A is the area of the coil (m²)
θ is the angle between the normal to the coil and the magnetic field (in radians)
The area of the coil can be calculated using the formula:
A = π * r²
Substituting the values:
A = π * (0.037 m)² ≈ 0.004315 m²
Next, we need to convert the angle from degrees to radians:
θ_rad = θ * π / 180
θ_rad = 45° * π / 180 ≈ 0.7854 radians
Now we can find the change in magnetic flux (ΔΦ):
ΔΦ = B * A * cos(0.7854)
Given that the average emf (ε) is 0.095 V, and we have one-eighth of a revolution within the time interval of 0.024 s, we can calculate the rate of change of magnetic flux (dΦ/dt):
dΦ/dt = ΔΦ / Δt
Now we can rearrange the formula for Faraday's Law to solve for the magnitude of the magnetic field (B):
B = -ε / (N * (dΦ/dt))
Substituting the known values:
B = -0.095 V / (660 turns * dΦ/dt)
Finally, you can calculate the magnitude of the magnetic field at the location of the coil by substituting the values and solving the equation.