Question

A RC circuit has an emf of $300\cos \left(2t\right)$ volts, a resistance of $150ohms$, a capacitance of $\frac{1}{600}farad$, and an initial charge on the capacitor of $5coulombs$. Find the charge on the capacitor at any time $t$ .

• A. $\frac{2}{5}(23e^{-4t}+\sin (2t)+2\cos (2t\left)\right)$
• B. $\frac{1}{5}(23e^{-4t}+\sin (2t)+2\cos (2t\left)\right)$
• C. $\frac{3}{5}(23e^{-4t}+\sin (2t)+2\cos (2t\left)\right)$
• D. $\frac{1}{6}(23e^{-4t}+\sin (2t)+2\cos (2t\left)\right)$

Answer 1

The correct option is B $\frac{1}{5}(23e^{-4t}+\sin (2t)+2\cos (2t\left)\right)$

The circuit equation is $E=\frac{dq}{dt}R+\frac{q}{C}$

Where $R$ is the resistance,$C$ is the capacitance,$E$ is the EMF supply and $q$ is the charge,

Substituting the given values gives,

$300\cos \left(2t\right)=150\frac{dq}{dt}+\frac{q}{600}$

$⟹2\cos \left(2t\right)=\frac{dq}{dt}+\frac{q}{90000}$

This is a Linear Differential equation in $q$,

The integrating factor is $e^{\frac{t}{90000}}$

$⟹q{e}^{\frac{t}{90000}}+c=\int 2e^{\frac{t}{90000}}\cos \left(2t\right)dt$

Given that Intital charge of th capacitor is $5coulombs$,

Which gives the value of integration constant $c$,

$\therefore$ charge on the capacitor at time $t$ is $\frac{1}{5}(23e^{-4t}+\sin \left(2t\right)+2\cos \left(2t\right))$