The Electric Field

An electric field is simply a map of the push a charge would feel at each spot around other charges. "Field" is the physicist's word for anything that has a value at every point in a region, like the temperature at every spot in a room. It's useful because you work out the map once, and then any charge that wanders in gets pushed according to it, no matter how much charge it has. The equation $\mathbf{F} = q\mathbf{E}$ is just two pieces: $\mathbf{E}$ (the map, telling you how hard and which way one coulomb would be pushed at that spot) and $q$ (how much charge you actually put there). Multiply them and you get the force. It sounds abstract, but it's only Coulomb's law split into two steps, and the rest of the lesson is just drawing and adding these maps.

From force to field. In the last lesson two charges pushed on each other across empty space, which is a little strange if you think about it. How does one charge "know" the other is there? The field picture splits the interaction into two steps. First, the source charges set up a field $\mathbf{E}$ at every point around them. Then any charge $q$ that shows up feels $\mathbf{F} = q\mathbf{E}$ from the field right where it is. To measure $\mathbf{E}$ at a point, put a small test charge $q_0$ there, measure the force on it and divide: $\mathbf{E} = \mathbf{F}/q_0$, in newtons per coulomb.

A point charge. Divide Coulomb's law by $q_0$ and you get $\mathbf{E} = \frac{kq}{r^2}\hat{\mathbf{r}}$: away from positive charges, toward negative ones, fading as $1/r^2$. Notice the test charge has dropped out completely. The field belongs to the sources alone, and that's the payoff: compute it once and you know the force on any charge you might put anywhere.

Field lines are curves drawn tangent to $\mathbf{E}$ everywhere:

• They start on positive charges and end on negative charges (or run off to infinity).
• The number of lines leaving or entering a charge is proportional to its charge.
• Where lines crowd together the field is strong; where they spread out it's weak.
• They never cross, because $\mathbf{E}$ has only one direction at each point.

Superposition. Forces from several charges add as vectors, and $\mathbf{E}$ is just force divided by $q_0$, so fields add the same way. The field of several charges is the sum of their separate fields: a vector field built one charge at a time.

The dipole. Equal and opposite charges $\pm q$ a distance $d$ apart form a dipole: a water molecule, the separated charge of a beating heart, a radio antenna. On its axis, a distance $z$ from the center,

Far away ($z \gg d$) the $d^2/4$ is negligible, leaving $E \approx 2kqd/z^3 = 2kp/z^3$. On the perpendicular bisector, a distance $y$ away, the components along the bisector cancel and the components parallel to the axis add: $E = \frac{kqd}{(y^2 + d^2/4)^{3/2}} \approx \frac{kp}{y^3}$, pointing from $+q$ toward $-q$.

Why one more power of distance? Think about what the dipole looks like from far away. The two charges almost sit on top of each other, so their fields almost cancel and only a small difference survives. The difference $\frac{1}{(z - d/2)^2} - \frac{1}{(z + d/2)^2}$ is about $-d$ times the slope of $1/z^2$, and that slope is $-2/z^3$, so the difference is about $2d/z^3$: a linear approximation. That extra power is why neutral molecules tug on each other far more weakly at a distance than ions do.

A dipole in a uniform field feels forces $q\mathbf{E}$ and $-q\mathbf{E}$. Those cancel, so the dipole doesn't go anywhere, but they act at different points, so they twist it: the torque is $\boldsymbol{\tau} = \mathbf{p}\times\mathbf{E}$ (a cross product), of size $pE\sin\theta$, turning $\mathbf{p}$ toward $\mathbf{E}$. A microwave oven reverses its field about 5 billion times a second; water molecules keep twisting to follow, and that jostling heats the food.

A common mistake. It's tempting to think a field line is the path a charge follows, since the line shows which way the charge gets pushed. It isn't. A charge released from rest starts off along its field line, but the field sets its acceleration, not its velocity, and its inertia carries it off the line wherever the line curves. A few other slips come up a lot. A charge isn't pushed by its own field, so leave it out when finding the force on that charge. $\mathbf{E}$ points toward a negative charge, and the force on a negative charge is opposite to $\mathbf{E}$. And fields pointing different ways add by components, never by magnitudes.

Capacitive touchscreens find your finger by how it disturbs the field between electrodes under the glass. Field mills at airports and launch pads measure the atmosphere's electric field to warn of lightning before it strikes, and high-voltage engineers compute the fields around insulators and connectors to keep them below air's breakdown strength of about 3 MV/m. Microwave ovens heat food by twisting water's molecular dipoles back and forth in a rapidly reversing field.