Summary

General Principles

Sources of E⃗

Electric fields are created by charges.

Multiple point charges

Model Model objects as point charges.

Visualize Establish a coordinate system and draw field vectors.

Solve Use superposition: E⃗=E⃗1+E⃗2+E⃗3+⋯

Continuous distribution of charge

Model Model objects as simple shapes.

Visualize

  • Establish a coordinate system.

  • Divide the charge into small segments ΔQ.

  • Draw a field vector for one or two pieces of charge.

Solve

  • Find the field of each ΔQ.

  • Write E⃗ as the sum of the fields of all ΔQ. Don’t forget that it’s a vector sum; use components.

  • Use the charge density (λ or η) to replace ΔQ with an integration coordinate, then integrate.

Consequences of E⃗

The electric field exerts a force on a charged particle:

F⃗=qE⃗
The figure consists of two parts that illustrate the force vectors of positive and charges in an electric field.

The force causes acceleration:

a⃗=(q/m)E⃗

Trajectories of charged particles are calculated with kinematics.

The figure illustrates the torque exerted on a dipole in an electric field.

The electric field exerts a torque on a dipole:

τ=pE sin θ

The torque tends to align the dipoles with the field.

In a nonuniform electric field, a dipole has a net force in the direction of increasing field strength.

The figure illustrates a dipole in a non-uniform field.

Applications

Four Key Electric Field Models

Point charge with charge q

The figure illustrates the electric field due to a positive point charge.
E⃗=14πϵ0qr2r^

Infinite line of charge with linear charge density λ

The figure illustrates the electric field due to an infinitely long positively charged rod.
E⃗line=(14πϵ02| λ |r, { away if +toward if − )

Infinite plane of charge with surface charge density η

The figure illustrates a horizontally oriented rectangular plane with positive charges.
E⃗plane=(| η  |2ϵ0, { away if +toward if − )

Sphere of charge with total charge Q

Same as a point charge Q for r>R

The figure illustrates a positively charged sphere of radius R.

Electric dipole

The figure illustrates electric dipole moment on a horizontally oriented cylindrical surface.

The electric dipole moment is

p⃗=(qs, from negative to positive)

Field on axis: E⃗=14πϵ02p⃗r3

Field in bisecting plane: E⃗=−14πϵ0p⃗r3

Parallel-plate capacitor

The electric field inside an ideal capacitor is a uniform electric field:

E⃗=(Qϵ0A, from positive to negative)
The figure illustrates the electric field inside a parallel plate capacitor.

Terms and Notation