We can use Faraday's law of electromagnetic induction to find the induced current in the coil. Faraday's law states that the induced emf in a coil is equal to the negative of the rate of change of magnetic flux through the coil. The emf is related to the current through the coil by Ohm's law, V = IR, where V is the voltage, I is the current, and R is the resistance of the coil.
The magnetic field inside the coil is given by B = B0e^(-t), where B0 = 0.80 T and = 200 s^-1. The magnetic flux through the coil is given by Φ = NBA, where N is the number of turns, B is the magnetic field, A is the area of the coil, and Φ is the magnetic flux. The area of the coil is given by A = l × w, where l is the length of the coil and w is the width of the coil.
The resistance of the coil is given as 7.0 Ω.
(a) At t = 0.001 s:
The magnetic flux through the coil is given by Φ = NBA = NBlw. The rate of change of magnetic flux with time is given by dΦ/dt = NBA(-dB/dt) = NBlwB0e^(-t).
The induced emf in the coil is given by emf = -dΦ/dt = -NBlwB0e^(-t).
The induced current in the coil is given by I = emf/R = (-NBlwB0/R) e^(-t) = (-40 * 0.80 T * l * w / 7.0 Ω) e^(-t).
Substituting t = 0.001 s, we get:
I = (-40 * 0.80 T * l * w / 7.0 Ω) e^(-0.001) = 2.64 A
Therefore, the induced current in the coil at t = 0.001 s is 2.64 A.
(b) At t = 0.002 s:
Using the same method as above, we can find the induced current in the coil at t = 0.002 s:
I = (-40 * 0.80 T * l * w / 7.0 Ω) e^(-0.002) = 1.85 A
Therefore, the induced current in the coil at t = 0.002 s is 1.85 A.
(c) At t = 2.0 s:
At t = 2.0 s, the magnetic field inside the coil has decreased significantly, and the induced current will be much smaller. We can use the same method as above to find the induced current in the coil at t = 2.0 s:
I = (-40 * 0.80 T * l * w / 7.0 Ω) e^(-2.0) = 0.003 A
Therefore, the induced current in the coil at t = 2.0 s is 0.003 A, which is much smaller than the currents induced at earlier times.