The rotor in a certain electric motor is a flat, rectangular...

The rotor in a certain electric motor is a flat, rectangular coil with 80 turns of wire and dimensions 2.50 cm by 4.00 cm. The rotor rotates in a uniform magnetic field of 0.800 T. When the plane of the rotor is perpendicular to the direction of the magnetic field, the rotor carries a current of 10.0 mA. In this orientation, the magnetic moment of the rotor is directed opposite the magnetic field. The rotor then turns through one-half revolution. This process is repeated to cause the rotor to turn steadily at an angular speed of $3.60 \times 10^{3}$ rev/min. (a) Find the maximum torque acting on the rotor. (b) Find the peak power output of the motor. (c) Determine the amount of work performed by the magnetic field on the rotor in every full revolution. (d) What is the average power of the motor?

Key Concepts

Rotational Power and Work

In rotational dynamics, power is the rate at which work is done, given by the product of torque and angular velocity. Work done in a full rotation is related to the integral of torque over the angle, linking the concepts of torque, angular displacement, and energy transfer in rotating systems.

Angular Velocity Conversion

Angular velocity conversion is an important concept in rotational dynamics, where quantities given in revolutions per minute must be converted into radians per second for consistent use in calculations. This conversion is essential to correctly relate rotational motion to linear quantities like power.

Torque on a Magnetic Dipole

Torque arises when a magnetic dipole is placed in a magnetic field, with a magnitude given by the cross product of the dipole moment and the magnetic field. The maximum torque occurs when the dipole moment is perpendicular to the field, which is crucial for determining the rotational forces in devices like electric motors.

Magnetic Dipole Moment

This concept refers to the measure of the strength and orientation of a magnetic source, which in a current loop is determined by the product of the current, the number of turns, and the area of the loop. It is fundamental in understanding how the loop interacts with external magnetic fields, as it characterizes the loop as a magnetic dipole.

Transcript

00:02 We're told to have a rotor and an electric motor, and we're given the following quantities about the rotor.

00:10 Part a of the problem is asking us to find what maximum torque can act on the rotor.

00:20 And the equation for torque is equal to the number of turns in a coil of wire n times the area a, times the magnetic field b, and then times the current i.

00:42 And these are all values that are given to us by the problem.

00:47 So we can just plug them in.

00:49 So we plug in m and we plug in a, we plug in b, and then we plug in, and then this will give us a torque.

01:13 That is equal to 6 .40 times 10 to the negative 4 newton meters.

01:36 Now in part b, we need to calculate the peak power output, and we know that power is equal.

01:47 Power is equal to torque times the angular velocity.

01:59 So the torque we calculated in the previous part, so we can just plug in torque, which was 6 .4 times 10 to the negative 4 nanometers, and the angular velocity was given to us by the problem, and we just need to convert it to radiance per second.

02:23 So this is going to be 3 .60 times 10 to the 3 .3.

02:30 Revolutions per minute, and then to convert it to radiance per second, we do 2 pi, 2 pi radiance, divide by one revolution times one minute divided by 60 seconds.

02:53 And so this will give us a power that is equal to 0 .2 41 watts.

03:11 Now for part c, we need to calculate the amount of work performed by the magnetic field onto the rotor...

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