LR Circuits

Suppose that a solenoid is connected to a battery through a switch. Beginning at t = 0, when the switch is closed, the battery causes charges to move in the circuit.

A solenoid has inductance (L) and resistance (R), and each of these influence the current in the circuit. Assume that total resistance in the circuit, including the internal resistance of the battery, is represented by R. Similarly, L includes the self-inductance of the connecting wires.

As the current i(t) in the circuit increases (from i = 0 at t = 0), a self-induced emf ε = –L di/dt is produced in the inductance whose sense is opposite to the sense of the increasing current. This opposition to the increase in current prevents the current from rising abruptly.

If there been no inductance in the circuit, the current would have jumped immediately to the maximum value defined by ε₀/R. But due to an inductance coil in the circuit, the current rises gradually and reaches a steady state value of ε₀/R as t → τ. The time taken by the current to reach about two-third of its steady state value is equal to by L/R, which is called the inductive time constant of the circuit.

Greater the value of L, the larger is the back emf, and longer it takes the current to build up. This role of an inductance in an electrical circuit is somewhat similar to that of mass in mechanical systems.

The spark seen while turning off a switch connected to an electrical appliance such as a fan, computer, geyser or an iron, essentially arises due to back emf.