Question

# 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...

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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