In the circuit shown in Fig. `emf E` , a resistance `R`, and coil inductances `L_(1)` and `L_(2)` are known. The internal resistance of the source and the coil resistances are neglible. Find the steady-state currents in the coils after the swich `Sw` was shorted.

Explanation

The equations are,
`L_(1) (di_(1))/(dt) = xi - R (i_(1) + i_(2))`
Then, `(d)/(dt) (L_(1) i_(1) - L_(2) i_(2)) = 0`
or, `L_(1) i_(1) - L_(2) i_(2)` = constant
But initally at `t = 0, i_(1) = i_(2) = 0`
so constant must be zero and at all times,
`L_(1) i_(1) = L_(2) i_(2)`
In the final steady state, current must obviusly be `i_(1) + i_(2) = (xi)/(R)` . Thus is steady state,
`i_(1) = (xi L_(2))/(R (L_(1) + L_(2)))` and `i_(2) = (xi L_(1))/(R (L_(1) + L_(2)))`