A 40-turn, 4.0-cm-diameter coil with R=0.40Ω surrounds a 3.0-cm-diameter solenoid. The solenoid is 20 cm long and has 200 turns. The 60 Hz current through the solenoid is I=I0sin(2πft).
What is I0 if the maximum induced current in the coil is 0.20 A?
DATA:
SOLUTION
The current through the coil is defined as:
Where is defined as:
Where, is the magnetic flux through the solenoid, and can be written as follows:
But, because is in the direction of , so:
Being, the area of the solenoid, so:
Now, the magnetic field is given by:
But, is given by:
Therefore, replacing (6) into equation (5) we have:
Replacing (7) and (4) into equation (3):
Replacing (8) into equation (2) we have:
or
taking derivative:
or
Replacing (12) into equation (1) we have:
Now, the current through the coil is maximum when:
Therefore, equation (13) can be written as:
Isolating we obtain:
Finally, replacing values of DATA given into equation (15) we have:
Solving we obtain:
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