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Figure shows a circuit consisting of an ...

Figure shows a circuit consisting of an ideal cell, an inductor `L`, and a resistor `R`, connected in series. Let switch `S` be closed at `t = 0`. Suppose at `t = 0`, the current in the inductor is `i_(o)`, then find out the equation of current as a function of time.

Text Solution

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Let at an instant `t`, the current in the circuit is `i` which is increasing at the rate `di//dt`.
Writing `KVL` along the circuit, we have
`L(di)/(dt) = varepsilon -iR`
`rArr int_(i_(0))^(i) (di)/(varepsilon - iR) =int_(0)^(t) (dt)/(L)`
`rArr` In `((varepsilon - iR)/(varepsilon - i_(0) R)) = - (Rt)/(L) rArr varepsilon - iR = (varepsilon - i_(0)R)e ^(-Rt//L)`
`rArr i = (varepsilon - (varepsilon - i_(0)R) e^(-Rt//L))/(R )`
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