You have a wire of length L=1.00 m from which to make the square coil of a dc motor. The current

step 1 So I have a little setup here of our situation. We have a wire that we're given the total length

You have a wire of length L=1.00 m from which to make the...

You have a wire of length $L=1.00 \mathrm{m}$ from which to make the square coil of a dc motor. The current in the coil is $I=1.7 \mathrm{A},$ and the magnetic field of the motor has a magnitude of $B=0.34 \mathrm{T}$ . Find the maximum torque exerted on the coil when the wire is used to make a single-turn square coil and a two-turn square coil.

Key Concepts

Trade-Off with Multiple Turns

When constructing a multi-turn coil with a fixed total length of wire, there is a trade-off between the number of turns and the area of each individual turn. Increasing the number of turns boosts the magnetic dipole moment by a factor of N, but it decreases the area available for each turn, affecting the overall torque produced by the coil.

Area Determination for a Square Coil

For a coil of fixed wire length, the geometry of the loop—such as a square coil—determines its area. With a square, the side length is one-fourth of the total perimeter, making the area proportional to the square of the wire length. This concept is crucial when optimizing the magnetic moment under a fixed length constraint.

Magnetic Dipole Moment

The magnetic dipole moment, defined as ? = NIA (where N is the number of turns, I is the current, and A is the area of the loop), characterizes the strength of a coil’s magnetic source. It is a key parameter in determining the interaction between the coil and an external magnetic field.

Torque on a Current Loop

A current-carrying loop in a magnetic field experiences a torque given by ? = ?B sin?, where ? is the angle between the magnetic moment and the magnetic field. Maximum torque is achieved when the loop is oriented such that sin? equals 1, meaning the magnetic moment is perpendicular to the magnetic field.

Transcript

00:01 So i have a little setup here of our situation.

00:04 We have a wire that we're given the total length of that is made into a square and put into a magnetic field.

00:13 And it's a motor, so we have this axle running down the middle.

00:16 And then this is set to rotate when it's in the magnetic field, so it rotates around the axle.

00:29 Okay, so now we know the total length of this wire, and it's wrapped into a square.

00:35 Which means that each side of this square is one quarter l, or l divided by 4.

00:43 But because this magnetic field is parallel to two sides, we have this top side here and the bottom side are parallel to b, which means that it's not going to have any force in this direction, so it's not going to have any contribution to the torque, the total torque.

01:01 Because we know that magnetic field relative to current involves a sign of phi or theta in the forces case.

01:12 There's an angle involved either way, and it has to be perpendicular to each other.

01:16 If they're parallel, they don't contribute any force.

01:19 So our area that we need to take into consideration, we only have two sides.

01:29 So this is the hard part for this problem, is figuring out what we need to use for.

01:32 Our area.

01:33 So our equation for torque, which is what we're looking for, is number of turns times the current, times the area, times the magnetic field, sine of phi.

01:58 And we want when our torque is at its max, which means that we want our, we want our phi here to be 90 degrees, because that maximizes this equation.

02:13 Every other value of phi, less or other than 90 degrees.

02:19 So when phi is 90 degrees, we have sine of 1, which is the highest value for sign.

02:25 Every other value is going to be less than 1, which will lower the value of this total torque.

02:30 So when sine is 90, or when the theta, our phi here is 90 degrees, we have our maximum we can get, which is why i've drawn b, i've drawn b perpendicular to two sides.

02:44 It can only be max for a total of being perpendicular to a total of two sides.

02:49 So we could draw up and down and reverse this axle, but this direction works fine for our maximum torque, which is perpendicular to two sides...

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