Q.A $12\ \text{pF}$ capacitor is connected to a $50\ \text{V}$ battery. How much electrostatic energy is stored in the capacitor?

Concept: Capacitor Energy Storage — the energy stored in a capacitor depends on its capacitance and the square of the voltage across it.

Step 1: Write the formula for electrostatic energy stored in a capacitor:

Step 2: Substitute the given values. Capacitance $C = 12\ \text{pF} = 12 \times 10^{-12}\ \text{F}$, voltage $V = 50\ \text{V}$:

Step 3: Compute step by step:

$(50)^2 = 2500$ …

The key idea here is that a capacitor stores energy in the electric field between its plates. When you connect it to a battery, work is done to move charge onto the plates, and that work gets stored as electrostatic potential energy. The formula $U = \frac{1}{2} C V^2$ comes directly from the fact that the voltage across the capacitor builds up linearly as charge accumulates — the average voltage during charging is half the final voltage, so the energy is half the product of final charge and final voltage.

Let’s work through it step by step.

1. Identify the given quantities.

   The capacitance is $C = 12\ \text{pF}$. Remember, $1\ \text{pF} = 10^{-12}\ \text{F}$, so $C = 12 \times 10^{-12}\ \text{F}$.

   The voltage of the battery is $V = 50\ \text{V}$. This is the potential difference the battery maintains across the capacitor once fully charged.

2. Recall the formula for electrostatic energy stored in a capacitor.

   There are three equivalent forms, depending on which variables you know:

Since we have $C$ and $V$ directly, the first form is the most convenient.

3. Substitute the values.

First, square the voltage: $50^2 = 2500$.

Then multiply:

Simplify step by step:

$\frac{1}{2} \times 12 = 6$, so

Now $6 \times 2500 = 15000$, so

4. Express in scientific notation. $15000 = 1.5 \times 10^4$, so $U = 1.5 \times 10^4 \times 10^{-12} = 1.5 \times 10^{-8}\ \text{J}$ …

Method: Computing Electrostatic Energy Stored in a Capacitor

This method finds the energy stored in a capacitor from whichever pair of quantities ($Q$, $V$, $C$) is given.

Steps

Step 1: Recall the three equivalent energy formulas

All three come from the same integral (work done to build up charge against the rising voltage) — pick whichever form uses only the quantities already given, so you don't need an intermediate value.

Step 2: Match the given data to a form …