SOLUTION: An inductor in an electric circuit VL=10-.5t for a 2 H inductor. Find the current in the circuit after 20 seconds if the initial current was zero.

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Question 1099377: An inductor in an electric circuit VL=10-.5t for a 2 H inductor. Find the current in the circuit after 20 seconds if the initial current was zero.
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
+V%5BL%5D+=+L%2A%28di%2Fdt%29+
Which, after integration gives:
++i+=+%281%2FL%29+%2A+%28i%5B0%5D+%2B+int%28+V%28t%29%2C+dt%2C+0%2C++T+%29%29++
We have
+i%5B0%5D+=+0+ amps
+T+=+20+ sec
+V%28t%29+=+10-%281%2F2%29t+ volts, assumed
+L+=+2+ Henries
So
+i+=+%281%2F2%29+%2A%28+0+%2B+int%2810-%281%2F2%29t%2C+dt%2C+0%2C+20%29%29+
+i+=++%281%2F2%29+%2A%28++10t+-+%281%2F4%29t%5E2+%29+ evaluated at t=20 and t=0
++i+=+%281%2F2%29+%2A+%28%28200+-+%281%2F4%29%2820%5E2%29%29+-+%280+-+0%29%29+ for +t%3E=0+
+highlight%28+i+=+50%29+ amps
—
Below is a graph of i (y-axis) vs t (x-axis) (ignore the part to the left and below the origin, i[0] = 0).
It happens that i(t) is maximal at t=20sec, after that the current decays back down toward zero, and would go below zero after t=40sec IF the voltage source continued to obey 10-5t.