Problem 3:

1. Find the magnitude of the magnetic flux through the coil.
2. Now suppose the magnetic field is time-dependent (t=time) and is given by B = (1-0.4 t) k in units of Tesla. What is the magnitude of the current in the coil at time t=5 s?

1. The magnetic flux is given by

      ø_B = N B · ñ dA

   If the coil is in the x-y plane, then the normal to the surface ñ is in the k or -k direction. Thus the dot product is easy ->
   B · ñ = (0.5i + 0.7k) · k = 0.7 T (we are only concerned with the magnitude of the flux, so we don't need to worry about the possible minus sign). This is constant over the area of the coil, so the integral is just the total area = (0.5 m)² and the total flux is

      ø_B = (12)(0.7 T)( (0.5 m)²) = 6.60 Wb

2. This requires Faraday's law. The changing magnetic flux will induce an EMF which will cause a current in the coil. We have

     E = - dø_B/dt = - d/dt [1-0.4t](N r²) = 0.4 N r² = 0.4 (12) ( (0.5 m)²) = 3.77 V

   Using V = IR, the current in the coil will be

      I = E/R = 3.77V/5 = 0.754 A.

Problem 4
Test 3
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