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Chapter 23: Problem 14

Three point charges \(+q\) and a fourth, \(-\frac{1}{2} q,\) are assembled to form a square of side \(a\). Find an expression for the electrostatic energy of this charge distribution.

Short Answer

Expert verified
The total electrostatic energy of the charge distribution is \(U = 5k \cdot \frac{q^2}{a} - 2k \cdot \frac{q^2}{\sqrt{2a}}\).

Step by step solution

01

Identify the Charges

First, identify all the charges and their locations. We have three charges of \(+q\) and one charge of \(-\frac{1}{2}q\). Each is located at the corners of a square, with the negative charge opposite one of the positive charges.
02

Calculate the Energy between Charges

Using the formula for electrostatic potential energy (\(U = k \cdot \frac{q_1 \cdot q_2}{r}\)), calculate the energy for each pair of charges. There are six pairs of charges.
03

Identify the Distances

For each pair of charges, the distance separating them is either \(a\) (distance of square side length), \(\sqrt{2a}\) (diagonal distance), or \(2a\) (opposite vertices). Pay attention to the signs of the charges as the formula requires absolute values.
04

Sum All the Energies

Finally, add together all the energies calculated in step 2 to get the total electrostatic energy. This leads to the solution: \(U = 5k \cdot \frac{q^2}{a} - 2k \cdot \frac{q^2}{\sqrt{2a}}\).

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Most popular questions from this chapter

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