A flat circular coil with 192 turns, a radius of 2.35 x 10-2 m, and a resistance of 0.359 Ù 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 ÄB/Ät = 0.696 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.

Question

A flat circular coil with 192 turns, a radius of 2.35 x 10-2 m, and a resistance of 0.359 Ù 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 ÄB/Ät = 0.696 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 1

To find the magnitude of the magnetic field at the center of the coil that is produced by the induced current, we can use Faraday's law of electromagnetic induction. This law states that the induced electromotive force (emf) in a circuit is equal to the negative rate of change of the magnetic flux through the circuit.

The magnetic flux (Φ) through a coil can be calculated using the formula:

Φ = B * A

where B is the magnetic field perpendicular to the plane of the coil and A is the area of the coil.

In this case, the magnetic field is changing at a rate of ΔB/Δt = 0.696 T/s. The area of the coil can be calculated using the formula for the area of a circle:

A = π * r^2

where r is the radius of the coil.

Plugging in the values, we have:

A = π * (2.35 x 10^-2 m)^2

Using this value of A, we can calculate the change in magnetic flux ΔΦ:

ΔΦ = B * ΔA

Substituting the given values, we have:

ΔΦ = (0.696 T/s) * π * (2.35 x 10^-2 m)^2

Now, we can use Faraday's law to find the induced emf (ε):

ε = -ΔΦ/Δt

ε = -(0.696 T/s) * π * (2.35 x 10^-2 m)^2

Finally, since the coil has a resistance of 0.359 Ω, we can use Ohm's law to determine the current (I) induced in the coil:

I = ε/R

I = -[(0.696 T/s) * π * (2.35 x 10^-2 m)^2] / 0.359 Ω

Now that we know the current induced in the coil, we can calculate the magnetic field (B') produced by this current at the center of the coil. The magnetic field produced by a circular loop of current at its center can be calculated using Ampere's law:

B' = (μ₀ * I) / (2 * R)

where μ₀ is the permeability constant (4π x 10^-7 T*m/A) and R is the radius of the coil.

Using the values we've found, we have:

B' = (4π x 10^-7 T*m/A) * I / (2 * (2.35 x 10^-2 m))

Simplifying this expression will give us the magnitude of the magnetic field at the center of the coil that is produced by the induced current.