# Faraday's Law (Flux on one coil due to current through another Coil)

## Homework Statement There are a lot of numbers in this problem. Just about the only way to get it right is to work out each step symbolically first,and then plug numbers into the final symbolic result.

Two coils of wire are aligned with their axes along the z-axis,as shown in the diagram. Coil 1 is connected to a power supply and conventional current flows counter-clockwise through coil 1, as seen from the location of coil 2. Coil 2 is connected to a voltmeter. The distance between the centers of the coils is 0.14 m.
Coil 1 has N_1 = 565 turns of wire, and its radius is R_1 =0.07 m. The current through coil 1 is changing with time. At t=0 s, the current through coil 1 is I_0 = 15 A. At t=0.4 s,the current through coil 1 is I_0.4 =6 A.
Coil 2 has N_2 = 280 turns of wire, and its radius is R2 = 0.03 m.

Inside coil 2, what is the direction of – d/dt during this interval?
+Z (correct)
What is the direction of the electric field inside the wire of coil 2, at a location on the top of coil 2?
-X(correct)
At time t=0, what is the magnetic flux through one turn of coil 2? Remember that all turns of coil 1 contribute to the magnetic field. Note also that the coils are not very far apart(compared to their radii), so you can't use an approximate formula here.
At t=0 Phi_1 turn = ??? T m2

***THIS IS WHERE I'M STUCK*** At t=0.4 s, what is the magnetic flux through one turn ofcoil 2?
At t=0.4 s Phi_1 turn = ??? T m2

What is the emf in one turn of coil 2 during this timeinterval?
|emf1 turn| = ??? V

The voltmeter is connected across all turns of coil 2. What is thereading on the voltmeter during this time interval?

During this interval, what is the magnitude of the non-Coulombelectric field inside the wire of coil 2? Remember that the emfmeasured by the voltmeter involves the entire length of the wiremaking up coil 2.
ENC = ??? V/m

At t=0.5 seconds, the current in coil 1 becomes constant, at 5 A. Which of the following statements are true?
1. The electric field inside the wire of coil 2 now points in the opposite direction.
2. The voltmeter now reads 0 V.
3. The voltmeter reading is about the same as it was at t=0.4 seconds.
4. The electric field inside the wire of coil 2 is now 0 V/m.

## Homework Equations

I = dV/R
dV= Emf
Emf= -N*dPhi_mag/dt = -N*(d/dt)(B*n*dA)
Phi = B*n*dA

N=number of turns, n = nhat normal unit vector to area.
R = resistance

## The Attempt at a Solution

If I had a resistance I know I could find emf and from there find the flux phi, but without it I'm stuck and don't know what to do.

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At time t=0, what is the magnetic flux through one turn of coil 2? Remember that all turns of coil 1 contribute to the magnetic field. Note also that the coils are not very far apart(compared to their radii), so you can't use an approximate formula here.
At t=0 Phi_1 turn = ??? T m2

***THIS IS WHERE I'M STUCK*** No dimensions are given on the coils so I suppose you will assume that the coils have no width in the z direction.

It looks like you are going to have to calculate the B field due to coil 1 and integrate that at coil 2 to find the flux at coil 2. Do you remember what sort of field a current in a loop generates?

After pondering about this question for the better part of half an hour, while watching SC2 :), I was able to figure it all out! :D

for the first two parts where it asks for the flux, phi.
I used the equation Phi = B*n.hat*dA
where B_loop = N * (μ_0/4*pi) * (2*pi*r^2*I) / (z^2+r^2)^(3/2)
where N, I, and r belong to the current carrying wire. and z is the distance between the center of both coils.
Then Phi = B * dA
where dA is the cross sectional area of coil 2.

Emf for a single loop was then found using Emf = -dPhi/dt
dPhi would be Phi(0.4) - Phi(0) and dt is 0.4 - 0

Emf for the whole loop was found by multiplying the previous answer times the number of loops in coil 2. Emf(tot) = Emf(single loop) * N

E_nc was found by using E_nc = Emf / (2*pi*r*N)
where E_nc is the non-column electric field. r is the radius of coil 2, and N is the number of turns in coil 2. 2*pi*r*N = the total length of the coil

Hope this helps other people!! :D