PX157 - B11 - field energy

total potential energy

U_{12} = \frac{q_{1} q_{2}}{4 π ϵ_{0} r}

the potential at $q_{1}$ due to $q_{2} : V_{1} (r) = \frac{q_{2}}{4 π ϵ_{0} e}$
the potential at $q_{2}$ due to $q_{1} : V_{2} (r) = \frac{q_{1}}{4 π ϵ_{0} e}$

\begin{aligned} ⟹ U_{12} & = \frac{q_{1} q_{2}}{4 π ϵ_{0} r} \\ = \frac{1}{2} \frac{q_{1} q_{2}}{4 π ϵ_{0} r} + \frac{1}{2} \frac{q_{1} q_{2}}{4 π ϵ_{0} r} \\ = \frac{1}{2} q_{1} V_{1} + \frac{1}{2} q_{2} V_{2} \end{aligned}

also true for $n$ charges:

\begin{aligned} U & = \sum_{i = 1}^{n} \frac{1}{2} q_{i} V_{i} \\ V_{i} & = \sum_{j = 1, j \neq i}^{n} \frac{U_{i j}}{q_{i}} \end{aligned}

eg: potential energy of pour point charges
$U = U_{12} + U_{13} + U_{14} + U_{23} + U_{24} + U_{34}$
- since $U_{i j} = u_{j i} :$

U = \frac{1}{2} (U_{12} + U_{13} + U_{14} + U_{21} + U_{23} + U_{24} + U_{31} + U_{32} + U_{34} + U_{41} + U_{42} + U_{43})

energy stored in a capacitor

U = \frac{1}{2} (+ Q) V_{+} + \frac{1}{2} (- Q) V_{-} = \frac{1}{2} Q (V_{+} - V_{-})

U = \frac{1}{2} Q V = \frac{1}{2} C V^{2} = \frac{1}{2} \frac{Q^{2}}{C}

recap: in a spring:

U = \frac{1}{2} k x^{2}

energy density of the electric field

energy density:

u = \frac{U}{V}

for parallel plate capacitor:

u = \frac{\frac{1}{2} C V^{2}}{A d}

also, $C = \frac{ϵ_{0} A}{d}$ , $V = E d :$

u = \frac{1}{2} ϵ_{0} E^{2}

also, $U = m c^{2}$ , $U = \frac{1}{2} ϵ_{0} E^{2} V$
eg: a capacitor is connected to a fully charged battery with potential, $V$ , that carries charge, $Q$ . it is then disconnected. the separation between the plates is doubled ( $d \to 2 d$ ). what happens to the associated quantities?

before	after

C_{0} = \frac{ϵ_{0} A}{d}

C_{n e w} = \frac{ϵ_{0} A}{2 d} = \frac{1}{2} C_{0}

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U_{0} = \frac{1}{2} \frac{Q^{2}}{C_{0}}

U_{n e w} = \frac{1}{2} \frac{Q^{2}}{C_{n e w}} = \frac{Q^{2}}{C_{0}} = 2 U_{0}

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V_{0} = V

V_{n e w} = \frac{U_{n e w}}{Q} = 2 V_{0}

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E_{0} = \frac{V_{0}}{d}

E_{n e w} = \frac{V_{n e w}}{2 d} = \frac{2 V}{2 d} = E_{0}

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u_{0} = \frac{1}{2} ϵ_{0} E_{0}^{2}

u_{n e w} = \frac{1}{2} ϵ_{0} E_{n e w}^{2} = u_{0}

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V o l u m e_{0} = A d

V o l u m e_{n e w} = 2 A d = 2 V o l u m e_{0}