A flat circular coil with 103 turns, a radius of 3.96 10-2 m, and a resistance of 0.456 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 = 0.704 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.

the answer is 0.00128T
can you show me the steps to get this answer? thank you

To find the magnitude of the magnetic field at the center of the coil, we can use Faraday's law of electromagnetic induction. According to the law, the induced electromotive force (EMF) in a closed loop is equal to the rate of change of magnetic flux through the loop.

The induced current can be calculated using Ohm's law: I = V/R, where I is the current, V is the induced EMF, and R is the resistance of the coil.

Now, let's break down the steps to find the magnitude of the magnetic field at the center of the coil:

1. Calculate the induced EMF (V):
We know that the induced EMF is equal to the rate of change of magnetic flux. Mathematically, this can be represented as:
V = -N * dΦ/dt
Where N is the number of turns in the coil, and dΦ/dt is the rate of change of magnetic flux.

Given that ΔB/Δt = 0.704 T/s, which is the rate of change of the external magnetic field, and the magnetic field is perpendicular to the plane of the coil, we can find the rate of change of magnetic flux as:
dΦ/dt = ΔB/Δt * A
Where A is the area of the coil. The area of a circular coil is given by A = π * r^2, where r is the radius of the coil.

Substituting the values:
dΦ/dt = (0.704 T/s) * (π * (3.96 * 10^(-2) m)^2)

2. Calculate the induced current (I) using Ohm's law:
We are given the resistance of the coil, which is 0.456 Ω. Using Ohm's law equation, I = V/R, we can find the induced current:
I = V / R

Substituting the values:
I = (-N * dΦ/dt) / R

3. Find the magnitude of the magnetic field (B) at the center of the coil:
The magnetic field at the center of a flat circular coil is given by:
B = μ₀ * (N * I) / (2 * R)
Where μ₀ is the permeability of free space, N is the number of turns, I is the current, and R is the radius of the coil.

Substituting the values:
B = μ₀ * (N * I) / (2 * R)

The permeability of free space, μ₀, is a constant value equal to 4π × 10^(-7) T·m/A. Substituting this value and the known values, we can calculate the magnitude of the magnetic field at the center of the coil (B).

After following these steps and performing the calculations, you should find that the magnitude of the magnetic field at the center of the coil is approximately 0.00128 T.