# Derivation Energy of the Electric Field

Level 4 (for physics pros)
Level 4 requires the knowledge of vector calculus, (multidimensional) differential and integral calculus. Suitable for advanced students.
Updated by Alexander Fufaev on

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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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