Answer to Question #159165 in Electric Circuits for Hillary

Question

A capacitor is charged through a resistor using a battery of constant EMF.
(i) Draw sketch graphs on the same time axis showing how the charge on the capacitor and how the current through the circuit vary with time. Qualitatively explain their shapes.

Answer 1

Explanations & Calculations

• The charge stored in a capacitor varies overtime under a given voltage.
• That dependence is expressed in the equation

$\qquad\qquad \begin{aligned} \small Q_{(t)}&= \small CV\Big[1-e^{-\frac{t}{RC}}\Big]\\ &= \small CV\bigg[1-\frac{1}{e^{\frac{t}{RC}}}\bigg] \end{aligned}$ $\scriptsize v= battery\,voltage,R=series\,resistance$

• According to this charge accumulated by the capacitor increases overtime & the the rate decreases as there exists an inverse exponential relationship.

• @$\qquad\qquad \begin{aligned} \small t=0&\implies\small Q_0=0\\ \small t\to\infty &\implies\small Q_{\infty}= CV \end{aligned}$ $\scriptsize charge\,increases\,and\,saturates : rate\,decreses$

• And at the very beginning, the effective resistance sensed by the circuit is that from the series resistor. Hence the startup current is

$\qquad\qquad \begin{aligned} \small V&= \small iR\\ \small i_0&= \small \frac{V}{R} \end{aligned}$

• According to the definition of current, it is $\small Q=it$. Plugging this into the capacitor's equation yeilds

$\qquad\qquad \begin{aligned} \small it= \small CV\bigg[1-\frac{1}{e^{\frac{t}{RC}}}\bigg]\\ \small i_{(t)}= \small CV.\frac{1}{t}\bigg[1-\frac{1}{e^{\frac{t}{RC}}}\bigg] \end{aligned}$ $\qquad\qquad\scriptsize t\to\infty\implies i_{\infty}=0$

• According to this drawn current decreases overtime exponentially.

• When this behavior is sketched, something like the following is given