Solving the equation and determining the current

General solution

The equation E = Ri + Ldi/dt has as its general solution the sum of a particular solution (steady state i_p = E/R) and a solution to the homogeneous equation i_h(t) = A*exp(-t/tau), where τ = L/R.

Therefore, i(t) = E/R + A*exp(-t/τ).

Initial condition

The current in a coil is a continuous function of time (continuity of magnetic energy). If i(0-) = 0 (circuit initially at rest), then i(0+) = 0, which implies A = -E/R.

Final expression

i(t) = (E/R) * (1 - exp(-t/tau))

u_L(t) = E * exp(-t/tau)

Graphical interpretation

At t = tau, the current reaches approximately 63 per cent of its final value E/R. After 5*tau, the system is considered to have reached steady state (i >= 99 per cent of E/R). The tangent to the i(t) curve at the origin intersects the asymptote i = E/R exactly at t = tau: this is the standard graphical method for measuring tau on an oscilloscope.

t	i(t)
0	0
tau	0.63*E/R
3*tau	0.95*E/R
5*tau	0.99*E/R

Common pitfall

Be careful with units: tau = L/R is expressed in seconds because [L] = ohm*s and [R] = ohm, so [L/R] = s. A common mistake is to reverse R/L instead of L/R.