A simple DC electric motor consists of a coil of width b, length L, and N turns, that is free to rotate in a uniform magnetic field B as shown in the diagram. A direct current I passes through the coil in the direction shown, via a “commutator” that ensures that the current changes direction every half rotation.
(a) Why do you need to change the direction of the current?
(b) In which direction does the coil rotate: clockwise or anti-clockwise?
(c) Derive an expression for the maximum force on the side L of the coil, and hence show that the maximum torque acting on the coil is given by

where A is the area of the coil.
(d) Draw a labelled quantitative graph of the torque as a function of time as the coil rotates, given that B = 0.56 T, N = 100 turns, I = 5.3 A , L = 0.2 m, b = 0.1 m and the coil rotates at 3000 RPM.
(e) As the coil rotates we have a changing magnetic flux passing through the coil, which generates a so-called “back emf”.
Why is it called a back emf?
Show that the maximum back emf is given by

where ? is the angular velocity of the coil, and calculate its value using the values given above.
(f) If the coil has a resistance of 7.3 ohm, what value of the source voltage would be required to run the motor?
(g) If the motor were required to turn a heavy load so that its speed dropped to 2000 RPM for the same source voltage, what qualitatively would happen to the back emf, the coil current and the torque?
What would happen to these quantities if the motor stalled completely i.e. it stopped rotating?
Based on these answers draw a rough sketch graph showing the torque on the vertical axis as a function of the speed of rotation on the horizontal axis.
Is this torque/speed characteristic a good one for use in an electric car? Explain.

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