A quick post about the voltage drop across a motor.

When we are dealing with some exercises on a physics textbook, sometimes we could run into a question with a motor. Under most circumstances, we would be asked to calculate the power of the motor with only a few conditions provided. According to the formula you can't be more famaliar with, the power could be easily yielded with P = IV applied. But why is not V zero if we assume the resistance of any piece of wire to be 0?

A motor can be regarded as a piece of wire winding maybe a piece of metal with permanent magnet surrounding it. Regardless all the stuff about magnet, consider only V = IR, no voltage drop should be made if we suppose the resistance of wire to be 0. Nonetheless, we can actually acquire the voltage drop across the motor if conditions are provided (e.g. the current, emf of a battery, resistance of other electrical components). What then causes the voltage drop?

fig. 1 A circuit with a resistor, a battery and a motor.

In fig. 1, we know that the current through the motor is 0.5A and the voltage drop is 1V according to Kirchhoff's voltage law (2 = 2I + 0.5/I). So the voltage drop does exist. This would be strange if we ignore the electromagnetic effect inside the motor. Mentioned before, the armature (coil + metal) could rotate due to the magnetic field inside the motor, which means that the rotation of the windings of the armature also cut the magnetic field lines, inducing electromotive force (emf) on account of Lenz's law (like a generator). Therefore, when an emf is applied to the motor, it would also produce it's own emf, called counter-emf or back-emf here since this induced emf reduces the current.

Back to fig. 1, if we remove the motor and connect only the battery and the resistor, the current will be 1A; the voltage drop across the resistor is 2V. Once we attach the motor and know that the power of it is 0.5W, then we can aquire the voltage drop across the motor (=1V). The voltage drop is then said to be the back-emf of the motor since the total emf is 1V as the drop across the resistor.

Whether the motor is ideal (the resistance of the windings inside is 0) or not, there is always a back-emf when an outside emf is applied.