Electric charge and its properties

A comb and a scrap of paper

Run a plastic comb through dry hair a few times. Hold it near some small bits of paper. The paper jumps up to the comb. Nothing touched the paper, and the comb is not a magnet. What pulled the paper?

The pull is an electric force, and it comes from electric charge. In this lesson we learn what charge is, how a body gets it, and the rules that charge always follows.

Two kinds of charge

Rub a glass rod with a silk cloth and hang it from a thread. Bring a second glass rod, rubbed in the same way, near the first. The rods push each other away. Now bring a plastic rod that was rubbed with fur. This time the glass rod and the plastic rod pull each other.

So there are two kinds of charge, and two simple rules:

• Like charges repel. Two glass rods push apart, and so do two plastic rods.
• Unlike charges attract. A glass rod and a plastic rod pull together.

Benjamin Franklin called the charge on the glass rod positive, and the other kind negative. We still use these names.

Where does charge come from?

Look inside an atom. The nucleus holds protons, which are positive, and neutrons, which have no charge. Electrons, which are negative, move round the nucleus. A proton has a charge of $+e$ and an electron has a charge of $-e$, where $e = 1.6 \times 10^{-19}$ C. A neutral atom has as many electrons as protons, so the charges cancel.

The protons stay locked in the nucleus. The outer electrons are held more loosely, and they can move from one body to another. So when two bodies are rubbed together, only electrons move.

• The body that gains electrons becomes negative.
• The body that loses electrons becomes positive.

Glass rubbed with silk gives some electrons to the silk. The glass becomes positive and the silk becomes negative. A plastic rod rubbed with fur takes electrons from the fur, so the rod becomes negative.

The unit of charge

The SI unit of charge is the coulomb (C). One coulomb is the charge that a current of one ampere carries in one second, so $1\ \text{C} = 1\ \text{A s}$.

One coulomb is a very large charge. It is $6.25 \times 10^{18}$ electrons. Rubbing gives charges that are far smaller, so we use small units:

• $1\ \text{mC} = 10^{-3}\ \text{C}$ (millicoulomb)
• $1\ \mu\text{C} = 10^{-6}\ \text{C}$ (microcoulomb)
• $1\ \text{nC} = 10^{-9}\ \text{C}$ (nanocoulomb)
• $1\ \text{pC} = 10^{-12}\ \text{C}$ (picocoulomb)

Conductors and insulators

Hold a copper rod in your hand and rub it with wool. It shows no charge. Do the same with a plastic rod, and it does. Why?

In a metal, each atom gives up one or more electrons that move freely through the whole body. A material like this is a conductor. Charge put on one end of a conductor spreads over the whole body. Through your hand it also flows into the earth, so the copper rod keeps nothing.

In plastic, glass, rubber and dry wood, the electrons are bound to their atoms. These are insulators. Charge put on an insulator stays where it was put. If the copper rod has an insulating handle, it can be charged by rubbing.

The earth is a huge conductor. It can take or give any number of electrons without a change in its own charge. Joining a body to the earth by a wire is called earthing.

Charging by contact

Touch a neutral metal sphere with a charged rod. Charge flows from the rod to the sphere, and the sphere ends with a charge of the same sign as the rod.

Charging by induction

A body can also be charged without touching it. This is induction, and the order of the steps matters.

1. Put a neutral metal sphere on an insulating stand.
2. Bring a negative rod near the sphere, without touching it. The free electrons in the sphere are pushed to the far side. The near side becomes positive. The total charge is still zero.
3. Touch the far side with a finger, or join it to the earth. Electrons flow out of the sphere into the earth.
4. Take the finger away first, and then take the rod away.

The sphere is left with a charge of the opposite sign to the rod. The rod does not lose any of its own charge. Charging by contact gives the same sign. Charging by induction gives the opposite sign.

Why does a charged comb pull a neutral scrap of paper?

Paper is an insulator, so its charges cannot flow. But they can shift a little inside each atom. When a negative comb comes near, the positive parts of the paper move slightly closer and the negative parts move slightly away. The nearer charges feel a stronger force than the farther ones. So the pull wins over the push, and the paper moves to the comb. In the next lesson we will see why a nearer charge pulls harder.

The gold-leaf electroscope

An electroscope shows whether a body is charged. It has a metal rod with two thin gold leaves hanging at the bottom, inside a glass jar. When you touch the knob at the top with a charged body, charge runs down to the two leaves. Both leaves have the same sign, so they repel and spread apart. More charge makes them open wider.

Three properties of charge

1. Charge is additive. The total charge of a system is the sum of its charges, with their signs. A body with $+7\ \mu\text{C}$ and a body with $-2\ \mu\text{C}$ together have $+5\ \mu\text{C}$. Charge is a scalar, so we add it as ordinary numbers.

2. Charge is conserved. In an isolated system the total charge never changes. Charge is not made and not destroyed. It only moves from one place to another. When glass is rubbed with silk, the glass gets $+Q$ and the silk gets $-Q$. The total was zero before, and it is zero after. This holds in nuclear changes too. A neutron has no charge, and when it decays it makes a proton and an electron: $0 = (+e) + (-e)$.

3. Charge is quantised. Any charge that we can measure is a whole number of electron charges:

A body can have $3e$ or $10^{12}e$, but never $2.5e$. The smallest charge that can be found alone is $e$. Robert Millikan showed this with tiny oil drops. Every drop he measured carried a whole number of times $e$.

Then why does charge seem to flow smoothly, like water? Because $e$ is so tiny. A charge of just $1$ nC is $6.25 \times 10^{9}$ electrons. Adding one more electron changes it by one part in six billion. We can never notice the steps, in the same way that we cannot see single grains in a heap of sand from far away.

Quarks, the particles inside a proton, have charges of $\pm e/3$ and $\pm 2e/3$. But quarks are never found alone, so the smallest charge that we can use is still $e$. Also, the charge of a body does not depend on its speed. A fast electron has the same charge as a slow one.

A worked example

A plastic comb is rubbed with wool and gains $2.5 \times 10^{11}$ electrons. Find its charge, and the mass that it gains. The mass of an electron is $m = 9.1 \times 10^{-31}$ kg, and the comb has a mass of $10$ g.

1. The charge. $Q = Ne = 2.5 \times 10^{11} \times 1.6 \times 10^{-19}\ \text{C} = 4.0 \times 10^{-8}\ \text{C} = 40\ \text{nC}$. The comb gained electrons, so the charge is $-40$ nC.
2. The mass. $\Delta m = N m = 2.5 \times 10^{11} \times 9.1 \times 10^{-31}\ \text{kg} = 2.3 \times 10^{-19}\ \text{kg}$.
3. Compare. The comb has a mass of $0.010$ kg. The extra mass is $2.3 \times 10^{-17}$ of it. No balance can show this, so in problems we ignore the mass of the electrons that move.

Sharing charge. Two identical metal spheres have $+12\ \mu\text{C}$ and $-4\ \mu\text{C}$. They touch, and then they are pulled apart. The total is $+8\ \mu\text{C}$. The spheres are identical, so the charge spreads equally, and each sphere has $+4\ \mu\text{C}$. Now one of them touches a third identical sphere that is neutral. They share $+4\ \mu\text{C}$, so each has $+2\ \mu\text{C}$. The spheres now hold $+2\ \mu\text{C}$, $+4\ \mu\text{C}$ and $+2\ \mu\text{C}$. The total is still $+8\ \mu\text{C}$, as conservation says.

Slide to $+400$. About how many billion electrons has the rod gained? Look at the change of mass. How many electrons would the rod need for a change of mass of even one gram?

Practise with new numbers

Try it in Python

The program gives a rod more and more electrons. It finds the charge, and the extra mass of the rod, for each number.

Common mistakes

• Saying that protons move when we rub two bodies. Protons stay in the nucleus. Only electrons move.
• Thinking that a body that gains electrons becomes positive. Electrons are negative, so a body that gains them becomes negative.
• Saying that charging by induction needs a touch. The charged rod never touches the sphere. Only the earth wire touches it.
• Taking away the rod before the earth wire. If the rod goes first, the electrons spread out again and flow back through the wire, and the sphere ends neutral.
• Forgetting that charge is conserved. Charge is moved from one body to another. It is never made.
• Mixing up units. $1\ \mu\text{C} = 10^{-6}$ C and $1\ \text{nC} = 10^{-9}$ C.

Remember these

• Like charges repel. Unlike charges attract.
• Only electrons move in rubbing. The body that gains electrons is negative.
• Unit: the coulomb, $1\ \text{C} = 1\ \text{A s}$. The electron charge is $e = 1.6 \times 10^{-19}$ C.
• Contact gives the same sign. Induction gives the opposite sign.
• Charge is additive, conserved and quantised: $Q = ne$.
• Identical metal spheres that touch share the total charge equally.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How many electrons must be taken from a neutral rod to give it a charge of $+3.2\ \mu\text{C}$?

2. A student claims to have found a charge of $2.4 \times 10^{-19}$ C on a tiny drop. Is this possible?

3. A glass rod is rubbed with silk, and the rod becomes positive. What is the charge of the silk?

4. A negative rod is brought near a neutral metal ball on an insulating stand, but it does not touch the ball. Is the ball charged? Where are the charges on it?