A flat circular coil with 113 turns, a radius of 3.64 10-2 m, and a resistance of 0.524 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 = 0.774 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.
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.
Faraday's Law states that the induced electromotive force (EMF) in a closed loop is equal to the rate of change of magnetic flux through the loop. Mathematically, it can be expressed as:
EMF = -N * (ΔΦ/Δt)
Where:
EMF is the induced electromotive force
N is the number of turns in the coil
(ΔΦ/Δt) is the rate of change of magnetic flux
In our case, we are interested in finding the magnitude of the magnetic field, so we rearrange the equation as follows:
(ΔΦ/Δt) = -EMF / N
The magnitude of the magnetic field at the center of the coil is given by:
B = (μ₀ / 2π) * (ΔΦ/Δt) / R
Where:
μ₀ is the permeability of free space, with a value of approximately 4π × 10⁻⁷ T m/A
R is the radius of the coil
Substituting the values given in the problem:
N = 113 (number of turns)
(ΔΦ/Δt) = 0.774 T/s (rate of change of magnetic field)
R = 3.64 × 10⁻² m (radius of the coil)
First, we can calculate the value of ΔΦ/Δt:
ΔΦ/Δt = -EMF / N
= -0.774 T/s / 113
≈ -6.84 × 10⁻³ T/s
Now, we can calculate the magnitude of the magnetic field at the center of the coil:
B = (μ₀ / 2π) * (ΔΦ/Δt) / R
= (4π × 10⁻⁷ T m/A / (2π)) * (-6.84 × 10⁻³ T/s) / (3.64 × 10⁻² m)
≈ -5.96 × 10⁻⁴ T
Therefore, the magnitude of the magnetic field at the center of the coil that is produced by the induced current is approximately 5.96 × 10⁻⁴ T.