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Question
The equation of electromotive force for an electric circuit containing resistance and self-inductance is E = `"Ri" + "L" "di"/"dt"`, where E is the electromotive force is given to the circuit, R the resistance and L, the coefficient of induction. Find the current i at time t when E = 0
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Solution
By voltage law
`"Ri" + "L" "i"/"dt"` = E
÷ By L, `"Ri"/"L" + "di"/"dt" = "E"/"L"`
`"di"/"t" + "R"/"L" "i" = "E"/"L"` ........(1)
This is a linear differential equation.
Integrating factor I.F = `"e"^(int "R"/"L" "dt")`
= `"e"^("R"/"L" "t")`
Its solution is given by
`"i" "e"^("R"/"L" "i") = int "E"/"L" "e"^("R"/"L" "t") "dt" + "c"`
`"i" "e"^("R"/"L" "i") = "E"/"L" xx "L"/"R" "e"^("R"/"L" "t") + "c"`
`"i" "e"^("R"/"L" "i") = "E"/"L" "e"^("R"/"L" "t") + "c"`
Divided by `"e"^("R"/"L" "t")`
i = `"E"/"R" + "e"^(- "Rt"/"L")`
At time t find i when E = 0
i = `0/12 + "Ce"^(- "Rt"/"L")`
i = `"Ce"^(- "Rt"/"L")`
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