Electric Potential at the Center of a Cube with Point Charges

Q1. What is the electric potential at the center of a cube of side-length 10 cm, if 8 point charges of 100 nC each are placed at the corners? The potential is defined to be zero at infinity.

Background

Topic: Electrostatics – Electric Potential due to Point Charges

This question tests your understanding of how to calculate the electric potential at a point due to multiple point charges, using the principle of superposition. The charges are symmetrically arranged at the corners of a cube, and you are asked to find the potential at the geometric center.

Key Terms and Formulas

• Electric Potential due to a Point Charge: The potential $V$ at a distance $r$ from a point charge $q$ is given by:

$V = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}$

• $\varepsilon_0$ is the permittivity of free space ($8.854 \times 10^{-12} \ \mathrm{C}^2/\mathrm{N}\cdot\mathrm{m}^2$)

• For multiple charges, potentials add algebraically: $V_{\text{total}} = \sum_i V_i$

• All charges are the same and equidistant from the center, so the calculation simplifies.

Step-by-Step Guidance

1. First, calculate the distance from the center of the cube to any corner. For a cube of side $a$, this distance is $r = \frac{a\sqrt{3}}{2}$.

2. Convert the side length to meters: $a = 10\ \mathrm{cm} = 0.10\ \mathrm{m}$, so $r = \frac{0.10 \times \sqrt{3}}{2}$ meters.

3. Write the expression for the potential at the center due to one charge:

   $V_1 = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}$

4. Since there are 8 identical charges, the total potential is $V = 8 \times V_1$.

5. Substitute the values: $q = 100\ \mathrm{nC} = 100 \times 10^{-9}\ \mathrm{C}$, $\varepsilon_0 = 8.854 \times 10^{-12}\ \mathrm{C}^2/\mathrm{N}\cdot\mathrm{m}^2$, and $r$ as calculated above.

Try solving on your own before revealing the answer!

Final Answer: $8.3 \times 10^4$ Volts

By plugging in all the values and performing the calculation, you find the potential at the center is $8.3 \times 10^4$ V. This result comes from summing the contributions from all 8 charges, each at the same distance from the center.