Summary

General Principles

Connecting V and $\vec{E}$

The electric potential and the electric field are two different perspectives of how source charges alter the space around them. V and $\vec{E}$ are related by

Δ V = V_{f} - V_{i} = - \int_{s_{i}}^{s_{f}} E_{s} d s

where s is measured from point i to point f and $E_{s}$ is the component of $\vec{E}$ parallel to the line of integration.

Graphically

\begin{array}{l} Δ V & = & {the negative of the area under the E}_{s} graph \\ E_{s} & = & - \frac{d V}{d s} \\ = & the negative of the slope of the potential graph \end{array}

The figure illustrates electric field versus distance graph.

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The Geometry of Potential and Field

The electric field

Is perpendicular to the equipotential surfaces.
Points “downhill” in the direction of decreasing V.
Is inversely proportional to the spacing $Δ s$ between the equipotential surfaces.

The figure illustrates the geometry of the electric potential and the electric field.

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Conservation of Energy

The sum of all potential differences around a closed path is zero.

∑ {(Δ V)}_{i} = 0

The figure illustrates the sum of potential differences of four points arranged in the form of a quadrilateral.

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Important Concepts

A battery is a source of potential.

The charge escalator in a battery uses chemical reactions to move charges from the negative terminal to the positive terminal:

Δ V_{bat} = ε

where the emf $ε$ is the work per charge done by the charge escalator.

The figure illustrates the charge escalator model of a battery.

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For a conductor in electrostatic equilibrium

The interior electric field is zero.
The exterior electric field is perpendicular to the surface.
The surface is an equipotential.
The interior is at the same potential as the surface.

The figure illustrates the properties a conductor in static equilibrium.

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Applications

Capacitors

The capacitance of two conductors charged to $\pm Q$ is

C = \frac{Q}{Δ V_{C}}

A parallel-plate capacitor has

C = \frac{ϵ_{0} A}{d}

The figure illustrates two parallel plates placed at a certain distance from each other vertically.

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Filling the space between the plates with a dielectric of dielectric constant $κ$ increases the capacitance to $C = κ C_{0}$ .

The energy stored in a capacitor is $u_{C} = \frac{1}{2} C (Δ V_{C})^{2}$ .

This energy is stored in the electric field at density $u_{E} = \frac{1}{2} κ ϵ_{0} E^{2}$ .

Combinations of Capacitors

Series capacitors

The figure illustrates three capacitors connected in series.

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C_{eq} = {(\frac{1}{C_{1}} + \frac{1}{C_{2}} + \frac{1}{C_{3}} + \dots)}^{- 1}

Parallel capacitors

The figure illustrates three capacitors connected in parallel.

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C_{eq} = C_{1} + C_{2} + C_{3} + \dots

Terms and Notation