A circular coil (980 turns, radius = 0.078 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 = 0.027 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 0.058 V is induced in the coil. Find the magnitude of the magnetic field at the location of the coil.
To find the magnitude of the magnetic field at the location of the coil, we can use Faraday's law of electromagnetic induction, which states that the induced electromotive force (emf) in a closed loop is equal to the rate of change of magnetic flux through the loop.
The formula for the induced emf is given by:
emf = N * ΔΦ / Δt,
Where,
emf is the induced electromotive force,
N is the number of turns in the coil,
ΔΦ is the change in magnetic flux through the coil, and
Δt is the change in time.
In this case, the normal to the coil was initially perpendicular to the magnetic field, and after a time interval of 0.027 s, it makes an angle of 45 degrees with the field. This means the normal has rotated by one-eighth of a revolution.
Since the coil is circular, it undergoes a change in area as it rotates. The change in magnetic flux is given by:
ΔΦ = B * ΔA,
Where,
ΔΦ is the change in magnetic flux,
B is the magnetic field strength, and
ΔA is the change in area.
The change in area ΔA can be calculated using the formula for the area of a circle:
ΔA = π * (r² - r₀²),
Where,
ΔA is the change in area,
r is the final radius of the rotating coil,
and r₀ is the initial radius of the rotating coil.
Substituting the values and solving for B, we get:
B = (emf * Δt) / (N * ΔA).
Let's calculate the magnitude of the magnetic field:
Given:
emf = 0.058 V,
N = 980 turns,
Δt = 0.027 s,
r = 0.078 m,
r₀ = 0 (since the initial radius is 0).
First, let's calculate the change in area:
ΔA = π * (r² - r₀²)
= π * (0.078² - 0²)
= 0.0192 m².
Now, we can calculate the magnetic field:
B = (emf * Δt) / (N * ΔA)
= (0.058 * 0.027) / (980 * 0.0192)
≈ 9.02 * 10⁻⁴ T.
Therefore, the magnitude of the magnetic field at the location of the coil is approximately 9.02 x 10⁻⁴ Tesla.
To find the magnitude of the magnetic field at the location of the coil, we can use the formula for the average induced emf in a rotating coil:
emf = N * B * A * ω * sin(θ)
Where:
- emf is the average induced emf (0.058 V)
- N is the number of turns in the coil (980 turns)
- B is the magnitude of the magnetic field
- A is the area of the coil (π * r^2, where r is the radius of the coil)
- ω is the angular velocity of the coil (2π / T, where T is the time for one-eighth of a revolution, 0.027 s)
- θ is the angle between the normal to the coil and the magnetic field (45°)
Let's substitute the given values into the formula and solve for B:
0.058 V = 980 * B * (π * (0.078 m)^2) * (2π / 0.027 s) * sin(45°)
Simplifying the equation:
0.058 V = 980 * B * 0.019164 m^2 * 234.4 s^(-1) * 0.7071
0.058 V = 325.364 B
B = 0.058 V / 325.364
B ≈ 0.000178 T
Therefore, the magnitude of the magnetic field at the location of the coil is approximately 0.000178 Tesla (T).