Question

A \(12\,pF\) capacitor is connected to a \(50\,V\) battery. The electrostatic energy stored in the capacitor is:

• A. \(2.5\times10^{-6}\,J\)
• B. \(3.5\times10^{-8}\,J\)
• C. \(2.5\times10^{-8}\,J\)
• D. \(1.5\times10^{-8}\,J\)

Hint:

Energy stored in a capacitor can be written as: \[ U=\frac{1}{2}CV^2 \] \[ U=\frac{Q^2}{2C} \] \[ U=\frac{1}{2}QV \] Use whichever form matches the given data.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When a capacitor is connected to a battery, work is performed to transfer electric charge from one plate to the other against the growing internal electric field. This electrical work is stored inside the dielectric space separating the plates as potential electrostatic field energy ($U$).
Step 2: Key Formula or Approach:
The electrostatic potential energy stored in a capacitor can be calculated using its capacitance ($C$) and the applied potential difference ($V$): $$ U = \frac{1}{2} C V^2 $$ Let's convert our given parameters into standard SI metric units: - Capacitance ($C$): $12 \text{ pF} = 12 \times 10^{-12} \text{ F}$ (since $1 \text{ picofarad} = 10^{-12} \text{ Farads}$) - Potential difference ($V$): $50 \text{ V}$
Step 3: Detailed Explanation:
Let's substitute our converted values directly into the energy formula: $$ U = \frac{1}{2} \times \left(12 \times 10^{-12} \text{ F}\right) \times (50 \text{ V})^2 $$ First, calculate the square of the potential difference value: $$ (50)^2 = 2500 $$ Substitute this back into our expression and simplify the multiplication coefficients: $$ U = \frac{1}{2} \times 12 \times 10^{-12} \times 2500 $$ $$ U = 6 \times 2500 \times 10^{-12} $$ $$ U = 15000 \times 10^{-12} \text{ J} $$ Let's adjust the scientific notation decimal placement to match the options: $$ U = 1.5 \times 10^4 \times 10^{-12} \text{ J} $$ $$ U = 1.5 \times 10^{4 - 12} \text{ J} = 1.5 \times 10^{-8} \text{ J} $$ This calculated result matches option (D).
Step 4: Final Answer:
The electrostatic energy stored in the capacitor is 1.5 $\times$ 10$^{-8}$ J.