Induced EMF in a small square loop of wire at center of an AC Circuit

[bw519]

Homework Statement

A simple circuit that is 0.3m wide and 0.2m tall is driven with an alternating current (AC) power supply and has a 5 ohm resistor. The AC supply applies a 20V amplitude at 60 Hz. What emf amplitude is induced in microvolts in a small 1cm square loop of wire at the center of the circuit? (you can assume the field is constant over the surface of the square)

Relevant Equations

Emf = NBAw

I used the voltage of the power supply and resistance to solve for the current in the larger circuit (20V/5ohms=4 amps). I am not sure if the equation listed above is the correct one I should be using, but I tried it using the following numbers. For omega, I used 2*pi*frequency. N should equal 1. For area I tried the area of the small square (0.01mx0.01m). I had more of an issue solving for B. I found equations for the magnetic field due to a circular loop and due to a long wire, but I had more trouble finding anything for a rectangular loop. Is this even the correct way to be solving this problem? If so, is there an equation for B that I should be using.

I found one video online that was solving for the B at the center of a square loop. I believe the equation was 4*(4*pi*10^-7)*I*sin(theta)/(4*pi*a/2) where a was the length of the sides of the square. I was trying to use that with the rectangular wire by summing two versions of that equation, using 2 as the first coefficient for two sides at a time that were the same length (instead of 4 all the same), different angles and the different lengths for the sides in place of a, but none of those options seemed to work.

 

[TSny]

I found one video online that was solving for the B at the center of a square loop. I believe the equation was 4*(4*pi*10^-7)*I*sin(theta)/(4*pi*a/2) where a was the length of the sides of the square. I was trying to use that with the rectangular wire by summing two versions of that equation, using 2 as the first coefficient for two sides at a time that were the same length (instead of 4 all the same), different angles and the different lengths for the sides in place of a, but none of those options seemed to work.

This approach should work. However, it appears that the formula you are trying to use might not be correct. Take a look at https://www.clutchprep.com/physics/practice-problems/144964/a-steady-current-i-is-flowing-through-a-straight-wire-of-finite-length-part-afin.

The question statement is not clear. Do they want the amplitude of the induced emf or the RMS average emf or maybe something else?

 

[rude man]

You can use Biot-Savart to determine the B field at the center of the large loop. Then emf = -dB/dt times area of small loop.