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 produced by the induced current, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the induced electromotive force (EMF) is equal to the rate of change of magnetic flux through the coil. We can then use Ohm's law to relate the EMF to the induced current and the resistance of the coil.
The first step is to find the magnitude of the induced EMF:
EMF = -N(dΦ/dt)
where N is the number of turns in the coil and dΦ/dt is the rate of change of magnetic flux.
The magnetic flux through the coil is given by:
Φ = B*A
where B is the magnetic field and A is the area of the coil. In this case, the coil is flat and circular, so the area is equal to the area of a circle with radius r:
A = πr^2
Taking the derivative of the flux with respect to time gives:
dΦ/dt = d(B*A)/dt = A*(dB/dt)
Now we can substitute this expression into the equation for the induced EMF:
EMF = -N(dΦ/dt) = -N*A*(dB/dt)
Finally, using Ohm's law, the induced EMF can be related to the induced current (I) and the resistance (R) of the coil:
EMF = I*R
Setting these two equations equal to each other:
-I*R = -N*A*(dB/dt)
Since we want to find the magnitude of the magnetic field (B) at the center of the coil, we can use the formula for the magnetic field produced by a current-carrying loop at its center:
B = μ0*N*I/(2*R)
Here, μ0 is the permeability of free space. To simplify the equation, we can substitute 2R for the expression for A:
B = μ0*N*I/R
Now we can substitute the expression for the induced current (I) into the equation for the magnetic field:
B = μ0*N*(-I*R)/(R)
B = -μ0*N*I
We know that the magnitude of the change in the magnetic field with respect to time (|dB/dt|) is 0.704 T/s. From the problem statement, we are given the number of turns (N = 103), the radius (r = 3.96 * 10^-2 m), and the resistance (R = 0.456 Ω).
To find the induced current (I), we can rearrange the equation for the induced EMF and solve for I:
I = EMF/R = -N*A*(dB/dt)/R
I = -N*A*(dB/dt)/R
I = -N*π*r^2*(dB/dt)/R
Substituting the given values into the equation:
I = -103*π*(3.96 * 10^-2)^2*(0.704 T/s)/0.456 Ω
Simplifying the expression:
I ≈ -0.00122 A
Finally, substituting the value of I into the equation for the magnetic field:
B = -μ0*N*I
B = -(4π * 10^-7 T*m/A)*103 * (-0.00122 A)
Simplifying the expression:
B ≈ 0.00128 T
Thus, the magnitude of the magnetic field at the center of the coil, produced by the induced current, is approximately 0.00128 T.