The electric field

Why fields are useful

Coulomb's law describes forces between charges, but it can be helpful to separate source charges from test charges. The electric field is a vector field that assigns an electric influence to every point in space. Once the field is known, the force on any charge placed there is simple to find.

The electric field is defined as electric force per unit positive test charge:

ec{E}=rac{ec{F}}{q}

Therefore, the force on a charge $q$ in an electric field is

ec{F}=qec{E}

If $q$ is positive, the force points in the direction of ec{E}. If $q$ is negative, the force points opposite ec{E}.

Field of a point charge

For a point charge $Q$, the electric field at distance $r$ has magnitude

E=k_erac{|Q|}{r^2}

In vector form,

ec{E}=k_erac{Q}{r^2}hat{r}

where $hat{r}$ points away from the source charge. If $Q$ is positive, the field points outward. If $Q$ is negative, the field points inward.

The field exists whether or not another charge is placed there. A test charge simply reveals the field by experiencing a force.

Superposition of fields

Electric fields obey superposition. The net electric field from multiple charges is

ec{E}_{net}=sum_i ec{E}_i

This is often easier than calculating forces directly because the field depends only on source charges, not on the test charge.

For point charges, calculate each contribution and add as vectors. In component form,

$E_x=sum_i E_{ix}$

$E_y=sum_i E_{iy}$

and similarly in three dimensions.

Field lines

Electric field lines are a visual tool. The tangent to a field line gives the direction of the electric field. The density of field lines represents field strength. Lines begin on positive charges and end on negative charges, or extend to infinity.

Field lines never cross. If they crossed, the field would have two directions at one point, which is impossible for a well-defined vector field.

Field lines are not physical strings. They are diagrams representing a vector field.

Continuous charge distributions

Real charge may be spread over lines, surfaces, or volumes. For continuous distributions, small charge elements $dq$ contribute small fields dec{E}:

dec{E}=k_erac{dq}{r^2}hat{r}

The total field is found by integration:

ec{E}=int dec{E}

Common charge densities include linear density

lambda=rac{dq}{dl}

surface density

sigma=rac{dq}{dA}

and volume density

ho=rac{dq}{dV}$$ Symmetry is essential for making these integrals manageable. ## Uniform field A nearly uniform electric field exists between large parallel plates with opposite charge. In a uniform field, the vector $ec{E}$ has the same magnitude and direction at many points. A charge in a uniform field experiences constant force and therefore constant acceleration if other forces are ignored: $$ec{a}=rac{qec{E}}{m}$$ This is similar to projectile motion in a uniform gravitational field, except the acceleration depends on charge sign and charge-to-mass ratio. ## Electric field and physical reality Historically, fields were introduced as a convenient calculation tool. In modern physics, fields are treated as physically real entities that carry energy and momentum. Electromagnetic waves are propagating electric and magnetic fields. ## The big idea The electric field describes how charges influence space. It is defined as force per unit positive charge, obeys superposition, and can be represented by field lines. Once the electric field is known, the force on any charge follows from $ec{F}=qec{E}$, making fields one of the central concepts of electromagnetism.