A flat circular coil with 97 turns, a radius of 4.04 10-2 m, and a resistance of 0.528 Ω is exposed to an external magnetic field that is directed perpendicular to the plane of the coil. The magnitude of the external magnetic field is changing at a rate of ΔB/Δt = 0.851 T/s, thereby inducing a current in the coil. Find the magnitude of the magnetic field at the center of the coil that is produced by the induced current.
Answer in T
To find the magnitude of the magnetic field at the center of the coil produced by the induced current, you can use Faraday's Law of Electromagnetic Induction. According to Faraday's Law, the induced electromotive force (emf) in a circuit is equal to the rate of change of magnetic flux through the circuit.
In this case, the induced emf is given by the expression:
emf = -N * (ΔΦ/Δt)
where N is the number of turns in the coil, and ΔΦ/Δt is the rate of change of magnetic flux through the coil with respect to time.
In a circular coil, the magnetic flux Φ through the coil is given by the expression:
Φ = B * A
where B is the magnetic field, and A is the area of the coil.
The area A of a circular coil is given by the expression:
A = π * r^2
where r is the radius of the coil.
Substituting these expressions into Faraday's Law, we get:
emf = -N * (Δ(B * A)/Δt)
Since the magnetic field is changing with time, we can write the expression as:
emf = -N * (B * ΔA/Δt + A * ΔB/Δt)
Since the coil is flat and the magnetic field is perpendicular to the plane of the coil, we can assume that the magnetic field is uniform throughout the coil. Thus, the magnetic field B is the same at all points in the coil.
At the center of the coil, the radius is equal to half the diameter of the coil. Therefore, the radius r is given by:
r = (1/2) * diameter
Since the diameter of the coil is twice the radius, we can substitute this expression to get:
r = (1/2) * 2 * r
Simplifying, we find that the radius r is equal to r.
Substituting into the expression for the area A, we get:
A = π * r^2
Differentiating both sides of this expression with respect to time, we get:
ΔA/Δt = 2π * r * (Δr/Δt)
Since the radius r is a constant, Δr/Δt is zero. Therefore, the rate of change of the area with respect to time is zero, and we can simplify the expression as:
ΔA/Δt = 0
Substituting this back into the expression for the induced emf, we get:
emf = -N * (B * ΔA/Δt + A * ΔB/Δt)
= -N * (0 + A * ΔB/Δt)
= -N * A * ΔB/Δt
Since the induced emf is equal to the resistance times the induced current, we can write the expression as:
emf = I * R
Substituting this into the previous expression, we get:
I * R = -N * A * ΔB/Δt
Solving for the induced current I, we get:
I = -(N * A * ΔB/Δt) / R
Plugging in the given values, we have:
N = 97
A = π * (0.0404 m)^2
ΔB/Δt = 0.851 T/s
R = 0.528 Ω
Substituting these values, we can now calculate the magnitude of the magnetic field at the center of the coil that is produced by the induced current using the equation:
B = -I * R / (N * A * ΔB/Δt)
Evaluate the expression using the given values and calculate the magnitude of the magnetic field at the center of the coil.