Derivation: Energy of the Electric Field

Positive charge - electric field lines

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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
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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
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Substitute the potential 1 into Eq. 2:

Infinitesimal energy using Coulomb potential
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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
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Let's integrate the right side:

Total energy of a charged sphere with uninserted integration limits
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Inserting the integration limits results:

Electrical energy of a charged sphere
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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:

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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
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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
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Distance \(d\) can be canceled once:

Energy via plate capacitor area and E-field
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Here \( A \, d\) is the volume \(V\) enclosed between the capacitor plates:

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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
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Note that...

All derived formulas for energy are valid only in vacuum. In order to make them also valid for E-fields in matter, they must be multiplied by the dielectric constant \(\varepsilon_{\text r}\) of the material.

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