In this problem I have to find the current through the section PQ of two loops of wire sharing one side.
The loop thing is in a uniform magnetic field pointing into the page which varies in based on the equation B=zt where t is in seconds and B is in teslas. z is a constant.
The wires in this problem have a resistance per length of y ohms/meter. y is a constant.
The circuit is composed of parts of length a. a is a constant.
|xxxxxxxxxxx|xxxxxx| *height is also a
In the above diagram, | and - represent wires. xxxxx represents the uniform magnetic field. .... is for spacing. the right loop is a square with side lengh a. The left loop is a rectange with one side length a and the other side length 2a.
area of a rectangle: side1 * side2
flux through a closed loop = magnetic field * area
emf = change in flux / change in time
The Attempt at a Solution
To start, I found the area of each loop.
area of loop 1 (larger loop) A_1 = 2*a*a =2a^2
area of loop 2 (smaller loop) A_2 = a*a = a^2
I then found the rate of change of flux which gives the emf:
(d\dt)(zt) * A so
for loop 1 the change in flux is zA_1 = z2a^2
for loop 2 the changfe in flux is zA_2 =za^2
I then determined the resistance in a segment of wire with length a:
which I can use to find the resistance in the sections of the loops that aren't shared.
This is 5R for loop 1 and 3R for loop 2.Also the resistance in the shared section is R.
I then used kirchoff's rules to come up with the equations and currents:
current 1 = I_1 which runs through loop 1.
current 2 = I_2 which runs through loop 2.
current 3 = I_3 which runs through the joined side of loop 1 and 2.
I1-I2 = I3 since the currents are flowing in opposite directions through PQ.
emf1 (change in flux for loop 1) - 5RI_1 - RI_3 = 0
emf2 (change in flux for loop 2) - 3RI_2 + RI_3 = 0 (+RI_3 because the current is flowing against the emf)
I then expanded, simplified etc which eventually got me
(3emf_1 -5emf_2)/7R = I_3 which is what I need to solve for except that my answer is wrong. I checked through my manipulations of the equations and I don't see the error but I can't rule out the possibility. So help with regards to where I went wrong in my approach/execution of this problem would be appreciated.