# Derivation Energy of the Electric Field

## Video - Maxwell Equations. Here You Get the Deepest Intuition!

Download video Unlock Let us consider a conducting sphere (e.g. a metal sphere) with **radius** \(r\). The sphere is charged step by step with small **charge portions** \(\text{d}Q\) coming from infinity. After the **total charge** \(Q\) is brought onto the sphere, the sphere produces the following **electric potential** (we assume this to be known here):

**Electric potential of a charge**

The **energy** \(\text{d}W\) of a charge portion \(\text{d}Q\) brought to the surface of the sphere with potential \(\phi_{\text e}(r) \) is:

**Infinitesimal energy depending on the potential at the location of the charge**

Substitute the potential 1

into Eq. 2

:

**Infinitesimal energy using Coulomb potential**

To get the total energy \(W_{\text{e}}\), we need to integrate the left side of Eq. 3

over the energy from 0 to \(W_{\text{e}}\) and integrate the right side over the charge from \(0\) to \(Q\):

**Integral for the total energy of a charged sphere**

W_{\text e} ~&=~ \frac{1}{4\pi\,\varepsilon_0 \, r} \, \int_0^Q Q \, \text{d}Q \end{align} $$

Let's integrate the right side:

**Total energy of a charged sphere with uninserted integration limits**

Inserting the integration limits results:

**Electrical energy of a charged sphere**

To eliminate the geometric factor, the radius \(r\), in 7

, we replace it with the **capacitance** \(C = 4\pi\,\varepsilon_0 \, r\) of a sphere:

**Electrical energy via charge**

Since we have eliminated the radius of the sphere and expressed the energy in terms of the capacitance \(C\), the equation 8

applies not only to a sphere but also to other charged bodies to which a capacitance can be assigned.

Since the charge \(Q\) is not so well accessible experimentally, we can express formula 9

by **voltage** \(U\). The definition of the capacitance \( C = Q/U \Leftrightarrow Q = C\,U\) is used for this purpose. The we insert \(Q\) in 8

:

**Electrical energy via voltage**

The assumption that the energy 9

is stored in the electric field can be motivated as follows: Consider a plate capacitor with **plate area** \(A\) and **plate distance** \(d\). Let \(U\) be the voltage between the capacitor plates. The **electric field** \(E\) inside the plate capacitor is given by \( E = U/d \Leftrightarrow U = E\,d\) and the capacitance by \(C = \varepsilon_0 A / d \). (see Derivation). Voltage and capacitance used in 9

results:

**Energy via plate capacitor area and E-field not simplified**

Distance \(d\) can be canceled once:

**Energy via plate capacitor area and E-field**

Here \( A \, d\) is the **volume** \(V\) enclosed between the capacitor plates:

**Electrical energy via E-field**

The relation 12

between the energy and the E-field motivates the assumption that the energy \(W_{\text e}\) is stored in the electric field \(E\), which is in the volume \(V\). The **energy density** \(w_{\text e} = W_{\text e}/V \) of the E-field is then:

**Energy density of the electric field**