Charge and the electric field

You shuffle across a carpet, reach for a doorknob, and a spark bites your finger. A balloon rubbed on your hair makes it stand up. Lightning is the same thing, scaled to the sky. What is the invisible stuff behind all of it, and why does it reach across empty space to grab you?

Start with what you can feel. Rubbing does something to objects that lets them pull or push on each other without touching. Two balloons rubbed the same way drift apart; the balloon and your hair pull together. There are clearly two flavours of this something, and same repels while opposite attracts. We call the something charge, and its two flavours positive and negative. That is the whole starting point: two kinds of charge, like pushes like away, opposites pull together.

Rubbing does not create charge. It moves charge. Some electrons, the light negative particles, get scraped off your hair and onto the balloon. The balloon ends up with extra electrons (negative); your hair, now short exactly that many, is left positive. Nothing was made. The books balance to the last electron. This is conservation of charge, and it is one of the most exact rules in all of physics: the total charge before equals the total after, always.^[1]

The field: play before the equation

Here is the part that feels like magic. Two charges push and pull across a gap with nothing in between. How does one charge know the other is there? The trick physics uses is to stop thinking of it as one charge reaching out to another. Instead, a charge fills the space around itself with an invisible influence, and that is what the second charge feels, right where it sits. Drag the charges below and watch the influence, drawn as little arrows, rearrange itself through the empty space instantly.

Each arrow shows the push a tiny positive test charge would feel if you placed it there. That map of pushes is the electric field. The field is not just where you can feel it - it exists at every point in the space, filled in whether or not a test charge is there to be pushed. When you release the green test charge, it does not compute distances to every source. It simply feels the arrow at its own location and follows it. The field is the local messenger.

How strong, and how it falls off

Experiment pins down the force between two charges precisely. It grows with each charge, and it weakens with the square of the distance between them. Written out, this is Coulomb's law (a stated experimental result^[2]):

You have met that shape before. If you have done the gravity lesson, this is the identical inverse-square form as Newton's $F=G{m}_1{m}_2/r^2$, with charge in place of mass. That is not a coincidence you have to accept blindly. Picture the influence spreading out from a point charge like paint sprayed evenly in all directions. By the distance $r$ it is smeared over the surface of a sphere, and a sphere's area is $4\pi r^2$. The same total influence spread over an area that grows as $r^2$ must thin out as $1/r^2$. Any influence that streams outward from a point and is conserved falls off this way, which is why gravity and electricity share it.

To strip out the test charge and describe only what the source does to space, we divide the force by the size of the test charge $q$. That defines the field:

And fields simply add. Two charges? At every point, add their two arrows tip to tail and you get the real field there. That vector sum is exactly what the widget computes, and it is exactly what your code would do.

import numpy as np

K = 8.99e9  # Coulomb constant, N m^2 / C^2

def field_at(point, charges):
    # charges: list of (q, position). Returns the net E vector at point.
    E = np.zeros(2)
    for q, pos in charges:
        d = point - pos
        r = np.linalg.norm(d)
        E += K * q * d / r**3   # (d / r^3) is the unit vector / r^2
    return E                    # a positive charge here feels force q * E

Why some things zap and others hold: conductors and insulators

Why does the shock jump from a metal doorknob but not a wooden one? In a conductor such as metal, some electrons roam freely, so charge flows and rearranges the instant a field appears. In an insulator such as rubber or wood, charges are pinned in place. The carpet charges you up; touch metal and all that excess charge dumps through the low-resistance path at once, a spark. That flowing charge is the subject of the next lesson.

Check yourself