A flat circular coil with 193 turns, a radius of 5.50 x 10-2 m, and a resistance of 0.241 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 = 0.908 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.
I did the following calculations, but still got the wrong answer. What am I doing wrong?
E=193(pi)(5.5E-2)^2(0.908)=1.6654
I=E/R=1.6654/0.241 ohms=6.9104 amps
B= (4pi E-7)(193) (6.9104) 2(5.5E-2)
B=1.8436 E-4
To find the magnitude of the magnetic field at the center of the coil produced by the induced current, we can use Ampere's Law.
Ampere's Law states that the line integral of the magnetic field around a closed loop is equal to the permeability of free space (μ₀) multiplied by the current passing through the loop.
In this case, the closed loop is the circular coil, and the current passing through it is the induced current due to the changing magnetic field.
The formula for the magnetic field produced by a current-carrying coil at its center is given by:
B = (μ₀ * N * I) / (2 * R)
Where:
B is the magnetic field at the center of the coil,
μ₀ is the permeability of free space (4π × 10^-7 T·m/A),
N is the number of turns in the coil,
I is the current passing through the coil,
R is the radius of the coil.
Let's calculate the magnetic field using this formula:
B = (4π × 10^-7 T·m/A) * 193 turns * 6.9104 A / (2 * 5.50 × 10^-2 m)
B = 1.8424 × 10^-4 T
Therefore, the magnitude of the magnetic field at the center of the coil is approximately 1.8424 × 10^-4 T.
If you got a different answer, there might be a calculation error or a mistake in the units used. Please double-check your calculations and ensure that you are using consistent units throughout.