Work and Energy

1. Work to Move a Charge

Based on the definition of electric potential, the work \(W\) required to move a test charge \(Q\) from point \(a\) to point \(b\) is simply the charge times the potential difference.

If we bring the charge in from infinity (where \(V=0\)) to a point \(\vec{r}\), the work done is:

2. The Energy of Point Charges

Imagine assembling a system of point charges from scratch.

• First charge (\(q_1\)): Takes 0 work (no field yet).
• Second charge (\(q_2\)): Must move against the field of \(q_1\). Work is \(q_2 V_1(\vec{r}_2)\).
• Third charge (\(q_3\)): Must move against fields of \(q_1\) and \(q_2\).

Summing the work for all charges gives the total energy of the system. To avoid double-counting pairs, we divide by 2:

Where \(V(\vec{r}_i)\) is the potential at the position of charge \(i\) caused by all other charges.

3. Continuous Charge Distributions

For a continuous distribution (like a charged cloud or solid object), the sum becomes an integral. We replace \(q_i\) with \(\rho d\tau\) and the summation with integration over the volume.

This equation represents the energy stored in the configuration of the charge density \(\rho\).

4. Where is the Energy Stored?

Using Gauss's Law (\(\rho = \epsilon_0 \nabla \cdot \vec{E}\)) and integration by parts, we can transform the equation above into one involving only the Electric Field.

This is a profound result. It suggests that the energy is not stored "in the charges," but in the electric field itself. Wherever there is a non-zero electric field, there is energy density.