Inductance

Draw the pattern of the magnetic field produced by electric current through a straight wire and through a wire coil:

Explain your answer using either the right-hand rule (conventional flow) or the left-hand rule (electron flow).

As an electric current is passed through a coil of wire, it creates a magnetic field. If the magnitude of this current changes over time, so will the strength of the magnetic field.

We also know that a magnetic field flux that changes over time will induce a voltage along the length of a wire coil. Explain how the complementary principles of electromagnetism and electromagnetic induction manifest themselves simultaneously in the same wire coil to produce self-induction.

Also, explain how Lenz’s Law relates to the polarity of the coil’s self-induced voltage.

In a simple resistor circuit, the current may be calculated by dividing applied voltage by resistance:

Although an analysis of this circuit probably seems trivial to you, I would like to encourage you to look at what is happening here from a fresh perspective. An important principle observed many times in the study of physics is that of equilibrium, where quantities naturally “seek” a state of balance. The balance sought by this simple circuit is equality of voltage: the voltage across the resistor must settle at the same value as the voltage output by the source:

If the resistor is viewed as a source of voltage seeking equilibrium with the voltage source, then current must converge at whatever value necessary to generate the necessary balancing voltage across the resistor, according to Ohm’s Law (V = IR). In other words, the resistor’s current achieves whatever magnitude it has to in order to generate a voltage drop equal to the voltage of the source.

This may seem like a strange way of analyzing such a simple circuit, with the resistor “seeking” to generate a voltage drop equal to the source, and current “magically” assuming whatever value it must to achieve that voltage equilibrium, but it is helpful in understanding other types of circuit elements.

For example, here we have a source of DC voltage connected to a large coil of wire through a switch. Assume that the wire coil has negligible resistance (0 Ω):

Like the resistor circuit, the coil will “seek” to achieve voltage equilibrium with the voltage source once the switch is closed. However, we know that the voltage induced in a coil is not directly proportional to current as it is with a resistor - instead, a coil’s voltage drop is proportional to the rate of change of magnetic flux over time as described by Faraday’s Law of electromagnetic induction:

Where,

v_coil = Instantaneous induced voltage, in volts

N = Number of turns in wire coil

\(\frac{dφ}{dt}\) = Instantaneous rate of change of magnetic flux, in webers per second

Assuming a linear relationship between coil current and magnetic flux (i.e. φ doubles when i doubles), describe this simple circuit’s current over time after the switch closes.