Question

Problem 10:   A conducting rod spans a gap of length L = 0.186 m and acts as the fourth side of a rectangular conducting loop, as shown in the figure. A constant magnetic field B = 0.65 T pointing into the paper is in the region. The rod is moving under an external force with an acceleration a = At2, where A = 6.5 m/s4. The resistance in the wire is R = 145 Ω. Randomized VariablesL = 0.186 m B = 0.65 T A = 6.5 m/s4 R = 145 Ω

13% Part (a) Express the magnitude of the magnetic flux going through the loop, Φ, in terms of B, x and L.
  13% Part (b) Express the speed of the rod, v, in terms of A and t. Assume v = 0 at t = 0.
  13% Part (c) Express the position of the rod, x, in terms of A and t. Assume x = 0 at t = 0.
  13% Part (d) Express the derivative of the magnetic flux, dΦ/dt, in terms of B, A, L and t.
  13% Part (e) Express the magnitude of the emf induced in the loop, ε, in terms of B, L, A and t.
  13% Part (f) Express the current induced in the loop, I, in terms of ε and R.
  13% Part (g) Express the current induced in the loop, I, in terms of B, L, A, t, and R.

  13% Part (h) Calculate the numerical value of I at t = 2s in A.

Answer 1

A)

= BA

where A is area enclosed

so,

= B * x * L

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B)

dv /dt = At²

dv = At² dt

integrate, we get

v = At³ / 3

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C)

dx = v dt

dx = ( At³ / 3)dt

integrate, we get

x = At⁴ / 12

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D)

magnetic flux, dΦ/dt = BLAt³ / 3

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e)

emf, = BLAt³ / 3

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F)

I = ​​​​​​​ / R

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G)

I = BLAt³ / 3R

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H)

I = 0.01445 A