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.
find induced EMF from 193 *area * dB/dt
E = (193)pi (5.5*10^-2)^2 (.908)
current = i = E/.241
now find B due to i
B = mu N i/2 r
= 4 pi*10^-7 (193)(i)/(2*5.5*10^-2)
I followed the above instructions but I kept getting the wrong answer. (My homework is online)
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
I redid the calculations multiple times and still got the wrong answer. Where am I going wrong?
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. According to Faraday's law, the induced electromotive force (emf) in a loop is equal to the rate of change of the magnetic flux through the loop.
The magnetic flux through the loop is given by the equation: Φ = B * A * cos(θ), where B is the magnetic field, A is the area of the loop, and θ is the angle between the magnetic field and the surface normal of the loop.
In this case, since the magnetic field is perpendicular to the plane of the coil, the angle θ will be 0 degrees. Therefore, cos(θ) will be equal to 1, simplifying the equation to: Φ = B * A.
The rate of change of the magnetic flux, dΦ/dt, is equal to the induced emf, ε. So we can write: ε = dΦ/dt = B * dA/dt.
Since the radius of the coil is given, we can calculate the area of the coil, A, by using the formula for the area of a circle: A = π * r^2.
The rate of change of the area, dA/dt, is zero for a static coil, so ε = B * 0 = 0. In this case, the induced emf in the coil will be zero.
Now let's calculate the current induced in the coil, using Ohm's law: V = I * R, where V is the voltage across the coil, I is the induced current, and R is the resistance of the coil.
Since the induced emf is zero, the voltage across the coil will also be zero. Therefore, I = V/R = 0/0.241 = 0 Amps.
Since there is no current in the coil, there will be no magnetic field produced at the center of the coil by the induced current. Therefore, the magnitude of the magnetic field at the center of the coil is zero.