An RC circuit has an emf given (in volts) by 400 cos 2t, a resistance of 100 ohms, and a capacitance of 10^-2 farad. Initially there is no charge on the capacitor. Find the current in the circuit at any time t.

Question

An RC circuit has an emf given (in volts) by 400 cos 2t, a resistance of 100 ohms, and a capacitance of [latex]10^{-2}[/latex] farad. Initially there is no charge on the capacitor. Find the current in the circuit at any time t.

Accepted Answer

We first find the charge q and then use (7.11) to obtain the current. Here, [latex]E=400 \cos 2 t, R=100[/latex], and [latex]C=10^{-2}[/latex] ; hence (7.10) becomes

[latex]\frac{d q}{d t}+q=4 \cos 2 t[/latex]

This equation is linear, and its solution is (two integrations by parts are required)

[latex]q=c e^{-t}+\frac{8}{5} \sin 2 t+\frac{4}{5} \cos 2 t[/latex]

At t = 0, q = 0; hence,

[latex]0=c e^{-(0)}+\frac{8}{5} \sin 2(0)+\frac{4}{5} \cos 2(0) \quad ext { or } \quad c=-\frac{4}{5}[/latex]

Thus [latex]q=-\frac{4}{5} e^{-t}+\frac{8}{5} \sin 2 t+\frac{4}{5} \cos 2 t[/latex]

and using (7.11), we obtain

[latex]I=\frac{d q}{d t}=\frac{4}{5} e^{-t}+\frac{16}{5} \cos 2 t-\frac{8}{5} \sin 2 t[/latex]

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Eq. (7.10): [latex]\frac{d q}{d t}+\frac{1}{R C} q=\frac{E}{R}[/latex]

Eq. (7.11): [latex]I=\frac{d q}{d t}[/latex]

Question 7.22: An RC circuit has an emf given (in volts) by 400 cos 2t, a r...

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