Question

The power supplied by a torque t to rotate an object to angular frequency w is p=tw. A common car engine produces t=300 Nm at w=3000 RPM (rotations per minute). Suppose you want to create an electric car that runs on battery and electric motor. Typically, electric car motors utilize a current of 100A and have a cros sectional area of 0.5 m2. Suppose also that the number of wire turns in the coil of the motor is N=100.

Q1. How strong of a magnetic field is required to produce the same torque as a regular car engine?

When the motor is connected in series with the battery that powers it, we can treat it like a generic circuit element with effective resistance R that causes a voltage drop V=IR. By setting the electrical power consumed by the motor equal to the mechanical power used to push the car, P=tw, the effective resistance of the motor can be determined.

Q2. What is the effective resistance of the motor?

Q3. Determine the voltage of the battery.

Answer 1

1) given N = 100
I = 100
A = 0.5
we know that
magnetic moment = NIA = 100*100*0.5 = 5e3
considering the magnetic field is at 90 to the loop i.e. maximum torque is generated
T = magnetic monent x B           (cross product)
T = magnetic moment * B * sin(theta)
300 = 5e3 * B * sin(90)
B = 0.06T

2) we know that
power = T*W
where w is in rad/sec
W = 2pi*3000/60 = 314rad/s
P = 300*314 = 94200
as stated
electrical power consumed = mechanical power
I^2*R = 94200
100^2 * R = 94200
R = 94200/10000 = 9.42ohm
3)
V = IR = 100*9.42 = 942V