Electrostatic Energy of Square Charge Configuration
Calculating work required to assemble four charges at square corners
Example 2.4
Four charges are arranged at the corners of a square ABCD of side d. (a) Find the work required to put together this arrangement. (b) Calculate extra work needed to bring a charge q₀ to the center.
Step 1: Bring +q to A
No work required (no other charges present)
W₁ = 0
Step 2: Bring -q to B
W₂ = -q × VB = -q²/(4πε₀d)
= -1.00 ×10⁻⁹ J
Step 3: Bring +q to C
W₃ = +q × (VC from A & B)
= -q²/(4πε₀d)(1-1/√2)
= -0.29 ×10⁻⁹ J
Step 4: Bring -q to D
W₄ = -q × (VD from A, B & C)
= -q²/(4πε₀d)(2-1/√2)
= -1.29 ×10⁻⁹ J
Total Work Required
Wtotal = W₁ + W₂ + W₃ + W₄ = -q²/(4πε₀d)(4-√2)
= -2.59 ×10⁻⁹ J
Key Observations
- The work done depends only on the final arrangement, not the assembly process
- This work equals the total electrostatic potential energy of the system
- Potential at center is zero (equal and opposite contributions cancel)
- No work is needed to bring a charge to the center
