Electric Fields

Unit Vector Notation

Using a unit vector ($\hat r$), we may rewrite the single point charge field equation (using the permittivity constant) of a source charge $q$ as

\[\vec E= \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat r\]

where $\hat r$ points from the charge to the point at which we wish to know the field. If positive, $\hat r$ points away from the point charge $q$ otherwise it points towards $q$.

Electric Field Models

Point Charge (Coulomb’s Law)

\[\vec E = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat r\]

or just the magnitude:

\[|\vec E| = \frac{kq}{r^2}\]

Line of Charge

\[\vec E = \left( \frac{1}{4\pi\epsilon_0}\frac{2|\lambda|}{r}=\frac{2kq}{lr}, \left\{\begin{array}{ll} \textrm{away if +}\\ \textrm{toward if -} \end{array}\right. \right)\]

where $\lambda$ = $\frac{q}{l}$ for some length $l$. Its units are $\frac{\textrm{C}}{\textrm{m}}$.

Plane of Charge

\[\vec E = \left( \frac{\eta}{2\epsilon_0}, \left\{\begin{array}{ll} \textrm{away if +}\\ \textrm{toward if -} \end{array}\right. \right)\]

where $\eta$ = $\frac{q}{A}$ for some area $A$. Its units are $\frac{\textrm{C}}{\textrm{m}^2}$.

Sphere of Charge

\[\vec E = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}\hat r\]

Electric Fields of Dipoles

We may represent an electric dipole with two opposite charges $\pm q$ separated by a small distance $s$.

\[(E_{dipole})_y \approx \frac{2kqs}{y^3}\]

gives the energy field of a dipole along the axis of the dipole, bisecting s. This only applies for $y\gg s$.

Electric Field Lines

Lines start on positive charges and end on negative charges
The closer together lines are, the greater the field strength
Lines never cross

Charge Densities

The linear charge density $\lambda$ for an object of length $L$ and total charge $Q$ is defined as

\[\lambda = \frac{Q}{L}\]

whose units are in $\frac{\textrm{C}}{\textrm{m}}$.

The surface charge density $\eta$ for an object with area $A$ and total charge $Q$ is defined as

\[\eta = \frac{Q}{A} = \frac{Q}{\pi R^2}\]

whose units are $\frac{\textrm{C}}{\textrm{m}^2}$

Electric Fields of Continuous Charge Distributions

Rod

For a rod of length $L$ and total charge $Q$, the electric field strength $E_{rod}$ at distance $r$ from the center of the charged rod is

\[E_{rod}=\frac{k|Q|}{r\sqrt{r^2+(L/2)^2}}\]

Suppose $r\gg L$, the rod’s electric field is the same as a point charge.

Infinite Rod

If the rod becomes very long line of charge while $\lambda$ remains constant,

\[E_{line}=\frac{2k|\lambda|}{r}=\frac{2kQ}{Lr}\]

Ring

For a ring of radius $R$ and total charge $Q$, the electric field strength $E_{ring}$ at distance $z$ on the ring’s axis is

\[E_{ring}=\frac{kzQ}{(z^2+R^2)^{3/2}}\]

Disk

For a disk of radius $R$ with surface charge density $\eta$, the on-axis electric field at distance $z$ is

\[E_{disk}= \frac{\eta}{2\epsilon_0}\left[1-\frac{z}{\sqrt{z^2+R^2}}\right]\]

Sheet

The electric field of a plane of charge with surface charge density $\eta$ is

\[E_{plane}= \frac{\eta}{2\epsilon_0}\]

Motion of Charged Particles within an Electric Field

A charged particle $q$ within an electric field experiences a force

\[\vec F_{\textrm{on q}} = q\vec E\]

And if $\vec F_{\textrm{on q}}$ is the only force acting on $q$ then it causes the particle to accelerate with

\[\vec a=\frac{\vec F_{\textrm{on q}}}{m}=\frac{q}{m}\vec E\]

The ratio $\frac{q}{m}$ is known as the charge-to-mass ratio. Two equal charges will experience equal forces however their acceleration will differ according to their mass. Two particles with the same charge-to-mass ratio will undergo the same acceleration and follow the same trajectory.

A particle in a uniform electric field (a field that is constant in magnitude and direction) will move with constant acceleration whose magnitude is

\[a=\frac{qE}{m}=\textrm{constant}\]

Motion of Dipoles within an Electric Field

Dipoles within a Uniform Electric Field

There is no net force on a dipole in a uniform electric field. Torque is applied by the field until the positive end of the dipole points in the same direction of the electric field, at which the dipole reaches equilibrium.

The torque on the dipole is

\[\tau = pE\sin(\theta)\]

where $p$ is the dipole moment $\vec p=qs$, and $s$ the distance between the the dipole ends.

Note that the torque is zero when the dipole is aligned with the field, at which $\theta=0$.

Alternatively, the torque may be defined as

\[\vec\tau = \vec p\times\vec E\]

Dipoles within a Non-Uniform Electric Field

The net force $\vec F_{net}$ on a dipole is in the direction of the strongest electric field
A dipole will experience a net force toward any charged object