Why Is Gravity So Weak?: Going Further 9–12

Charges push and pull

You learned in Grade 9 that an atom has a positive nucleus with negative electrons around it, and that charge is measured in coulombs (C): one electron carries a charge of 1.6 × 10⁻¹⁹ C Grade 9: The units of charge and current. Usually an object has exactly as many electrons as protons, and its charges cancel: it is neutral.

Rubbing can move electrons from one material to another. When you comb dry hair, electrons move from your hair onto the plastic comb. The comb ends up with extra electrons, so it is negatively charged; your hair is left short of electrons, so it is positively charged. No charge is made or destroyed: the comb gains exactly what your hair loses. This is the conservation of charge. Charge that stays put on an object like this is called static electricity.

Experiments with charged things show three rules:

There are two kinds of charge, which Benjamin Franklin named positive and negative in the 1740s.
Like charges repel: two negatively charged balloons push each other apart.
Unlike charges attract: the negative comb pulls on your positive hair, which is why your hair stands up towards it.

This is the first big difference from gravity. Gravity only ever pulls. The electric force can pull or push.

Activity

Static at home

You need: two balloons, thread, a plastic comb or ruler, small scraps of tissue paper, and a tap. It works best on a dry day.

Blow up both balloons and tie each to a thread about 50 cm long.
Rub one balloon on your hair (or a woolen jumper) for ten seconds, then hold its thread. Bring your hand near it, and then the place on your hair you rubbed.
Rub both balloons on your hair. Hold the two threads together in one hand so that the balloons hang side by side.
Comb your hair briskly, and hold the comb just above the paper scraps. Raise it slowly.
Turn on the tap so that a thin, smooth stream of water runs. Bring the charged comb close to the stream, without touching it.

What you should see

The balloon swings towards your hand: its charge pulls the opposite charge in your hand slightly nearer, and the near charge attracts it (the same thing happens with the paper in step 4). It swings even more strongly towards your hair: the balloon has taken electrons from your hair, so they have opposite charges, and attract.
The two balloons swing apart and hang at an angle: both have extra electrons, like charges, so they repel.
The scraps jump up to the comb. The paper is neutral, but the negative comb pushes the electrons in the paper slightly away from it (or pulls them closer, if the comb is positive). The near side of each scrap ends up with the opposite charge to the comb, and the far side the same charge. The near side is closer, so its attraction is stronger than the far side’s repulsion, and the scrap is pulled. (Sometimes a scrap touches the comb, picks up some of its charge, and then jumps away: now they have like charges.)
The stream bends towards the comb, whichever charge the comb has. Tap water conducts a little, so the comb draws charge of the opposite sign down the stream from the tap, and that charge is attracted. (Many books say instead that the water molecules turn to face the comb; that happens, but the charge drawn in from the tap does most of the bending.)

Long hair rising up from a person's head and clinging to a red balloon held just above it — **Figure 8.1** A balloon rubbed on hair takes electrons from it. The hair, now positive, is pulled towards the negative balloon, against gravity. Photo: MikeRun, CC BY-SA 4.0

You rub a balloon on a woolen jumper. Afterwards the balloon is negatively charged. What has happened?

Pick an answer to see why.

Coulomb’s law

How strong is the push or pull between two charges? In 1785 the French engineer Charles-Augustin de Coulomb measured it. He hung a light rod from a fine wire, with a small charged ball on one end, and brought a second charged ball near it. The force between the balls turned the rod and twisted the wire, and the twist told him the size of the force. This instrument is a torsion balance; you met the same idea in Grade 10, when Cavendish used one to measure gravity’s tiny pull Grade 10: The gravitational constant, G.

An old engraving of Coulomb's torsion balance: a wide glass cylinder with a scale round it, a tall tube rising from its lid to a knob at the top, and its parts drawn separately beside it — **Figure 8.2** Coulomb’s torsion balance, engraved for his paper of 1785. A fine silver wire runs down the tall tube and holds a light needle inside the glass case, with a small ball on one end. A second charged ball is lowered in through the lid. Turning the knob at the top twists the wire, and the scale round the case shows how far the needle has turned. Photo: Charles-Augustin de Coulomb, Public domain

Coulomb found:

Double the charge on one ball, and the force doubles. The force is proportional to the product of the charges.
Double the distance between them, and the force falls to one quarter. The force is inversely proportional to the square of the distance.

Does that sound familiar? It is exactly the pattern of Newton’s law of gravitation Grade 10: Newton’s universal law of gravitation, with charge in place of mass. Written as a formula, Coulomb’s law is

$F = \frac{k\,q_1 q_2}{r^2}$

where q₁ and q₂ are the two charges in coulombs, r is the distance between their centers in meters, and k is Coulomb’s constant:

$k = 8.99 \times 10^{9}\ \text{N m}^2/\text{C}^2$

(You may also see k written as $\dfrac{1}{4\pi\varepsilon_0}$ ; it is the same number.) Like gravitation, the force is mutual: each charge pushes or pulls the other with the same force, along the line joining them (Newton’s third law).

Put the signs of the charges in, and the sign of F tells you which way it acts. If both charges have the same sign, q₁q₂ is positive: a push (repulsion). If they have opposite signs, q₁q₂ is negative: a pull (attraction).

Change each charge, from negative through zero to positive, and the distance between the balls. Watch which way the arrows point, and how many times the force changes when you double a charge or double the distance.

Charge on ball 1: +20 nC Charge on ball 2: −20 nC Distance between centers: 10 cm

Force on each ball: 3.60 × 10⁻⁴ N
Direction: pulled together

The charges have opposite signs, so the balls pull each other together, each with the same force.

Static charges are usually tiny, so we measure them in smaller units: the microcoulomb (1 µC = 10⁻⁶ C) and the nanocoulomb (1 nC = 10⁻⁹ C). A balloon rubbed on hair picks up something like a tenth of a microcoulomb (it varies a lot from balloon to balloon: take this as the size, not an exact figure).

Worked example

Two balloons

Two rubbed balloons each carry a charge of −0.10 µC. Their centers are 30 cm apart. Treating each charge as if it were at the balloon’s center, what is the force between them, and is it a push or a pull?

Answer

Given: q₁ = q₂ = −0.10 µC = −1.0 × 10⁻⁷ C; r = 30 cm = 0.30 m.
$F = \frac{k q_1 q_2}{r^2} = \frac{8.99 \times 10^{9} \times (-1.0 \times 10^{-7}) \times (-1.0 \times 10^{-7})}{(0.30)^2}$
$F = \frac{8.99 \times 10^{9} \times 1.0 \times 10^{-14}}{0.090} = 1.0 \times 10^{-3}\ \text{N}$
F is positive (two negative charges), so it is a push: each balloon is pushed away from the other with about 1.0 × 10⁻³ N. That is about the weight of a tenth of a gram, enough to swing two light balloons a little apart on their threads.

Worked example

Inside a hydrogen atom

In a hydrogen atom, the electron is typically about 5.29 × 10⁻¹¹ m from the proton (this distance is called the Bohr radius). Calculate a. the electric force between them, and b. the gravitational force between them. (Charge of each: 1.60 × 10⁻¹⁹ C; mass of the proton 1.67 × 10⁻²⁷ kg, of the electron 9.11 × 10⁻³¹ kg; G = 6.67 × 10⁻¹¹ N m²/kg².) c. How many times stronger is the electric force?

Answer

First, the distance squared: r² = (5.29 × 10⁻¹¹)² = 2.80 × 10⁻²¹ m².
a. The proton is + and the electron −, so this is an attraction:
$F_e = \frac{k q_1 q_2}{r^2} = \frac{8.99 \times 10^{9} \times (1.60 \times 10^{-19})^2}{2.80 \times 10^{-21}} = \frac{2.30 \times 10^{-28}}{2.80 \times 10^{-21}} = 8.22 \times 10^{-8}\ \text{N}$
b.
$F_g = \frac{G m_1 m_2}{r^2} = \frac{6.67 \times 10^{-11} \times 1.67 \times 10^{-27} \times 9.11 \times 10^{-31}}{2.80 \times 10^{-21}} = \frac{1.01 \times 10^{-67}}{2.80 \times 10^{-21}} = 3.63 \times 10^{-47}\ \text{N}$
c.
$\frac{F_e}{F_g} = \frac{8.22 \times 10^{-8}}{3.63 \times 10^{-47}} = 2.3 \times 10^{39}$
The electric pull is about 2 × 10³⁹ times the gravitational pull. Inside an atom, gravity is far too weak to matter at all: it is the electric force that holds every atom together.

Two charges push each other apart with a force F. If both charges are tripled and the distance is kept the same, the force becomes

Pick an answer to see why.

A positive charge and a negative charge are 10 cm apart. What happens to the force between them if the distance is changed to 5 cm?

Pick an answer to see why.

Gravity against electricity

The two laws have exactly the same shape:

	Gravitation	Electric force
law	$F = \dfrac{G m_1 m_2}{r^2}$	$F = \dfrac{k q_1 q_2}{r^2}$
acts between	masses	charges
constant	G = 6.67 × 10⁻¹¹ N m²/kg²	k = 8.99 × 10⁹ N m²/C²
direction	always a pull	a pull (unlike charges) or a push (like charges)
reach	gets weaker as 1/r², never reaches zero	gets weaker as 1/r², never reaches zero
blocked by anything?	no	yes: a closed metal box shuts out electric forces

Look at the two constants. k is about 10²⁰ times as big as G. That alone doesn’t settle which force is stronger, because a coulomb and a kilogram are different things. A fair test is two real particles.

Worked example

Two protons

Compare the electric push and the gravitational pull between two protons, each of mass 1.67 × 10⁻²⁷ kg and charge 1.60 × 10⁻¹⁹ C, a distance r apart.

Answer

Divide one law by the other. The r² is the same in both, so it cancels:
$\frac{F_e}{F_g} = \frac{k q^2 / r^2}{G m^2 / r^2} = \frac{k q^2}{G m^2}$
$\frac{F_e}{F_g} = \frac{8.99 \times 10^{9} \times (1.60 \times 10^{-19})^2}{6.67 \times 10^{-11} \times (1.67 \times 10^{-27})^2} = \frac{2.30 \times 10^{-28}}{1.86 \times 10^{-64}} = 1.2 \times 10^{36}$
The electric push is about 1.2 × 10³⁶ times the gravitational pull, and because the distance canceled, that is true at every distance. Two electrons are even more lopsided: about 4 × 10⁴² times.

So why does gravity, so incredibly weak, hold the planets in their orbits, while the electric force seems to do nothing on that scale?

Because of the minus sign. Mass only comes one way, so every kilogram of the Earth pulls on you, and the pulls of all its 10⁵⁰ or so atoms add up. Charge comes in two kinds that cancel. The Earth has almost exactly as many electrons as protons, so the pull of its protons on any charge is canceled by the push of its electrons. A planet’s electric force is nearly zero, not because the force is weak, but because it is so perfectly balanced.

The strength of the electric force is exactly what keeps everything so balanced. If some part of a body had a few spare electrons, they would push one another apart and pull in positive charge from around them until the balance was restored. Lightning and the spark from a jumper are what happens when that rebalancing is sudden.

Pick a pair of bodies and give both the same tiny share of spare electrons. Compare the electric push with gravity’s pull. How small a share is enough to beat gravity? Then try the other pairs.

Pair:

Spare electrons on each: 1 in every 10²⁰

Gravity’s pull: 6.7 × 10⁻¹¹ N
Electric push: 2.1 × 10⁻¹⁵ N
Push ÷ pull: 3.1 × 10⁻⁵

Gravity wins: the bodies are so nearly neutral that the electric push is smaller than gravity’s pull. Move the slider right: how few spare electrons does it take to beat gravity?

You should find that the electric push overtakes gravity between one spare electron in 10¹⁸ (gravity still wins) and one in 10¹⁷ (the push wins): they balance at about one in 6 × 10¹⁷, and that the answer is the same for two balls, two people, or the Earth and the Moon. (The bigger the bodies, the more mass they have, but also the more electrons, in the same proportion.) Real planets are more neutral than that, so gravity wins.

Think about it

Why is gravity so weak?

We can measure how weak gravity is very precisely. But why is it so weak? Why is G so small, or a proton so light, that two protons pull each other 10³⁶ times more weakly than they push? Is there a reason, or is it just how our universe is?

Answer

Nobody knows. Physicists call it one of the great unsolved puzzles. Some have suggested that gravity “leaks” into extra dimensions of space we can’t see, but experiments such as the Large Hadron Collider have looked for signs of them and found none (as of 2026). Others think the numbers are simply the ones our universe happens to have. No answer has been confirmed. What we do know is that if gravity were much stronger, stars would be smaller and shorter-lived, and you would be crushed by your own weight: the weakness of gravity is part of why big, slow things like people and planets can exist.

Why does gravity, and not the electric force, hold the Moon in its orbit round the Earth?

Pick an answer to see why.

Fields: how a force reaches across space

The comb pulls the paper without touching it; the Earth pulls the Moon across 384,000 km of empty space. How does one thing push or pull another far away?

In the 1830s the English scientist Michael Faraday pictured it like this. A charge changes the space around it, even when nothing else is there. Bring a second charge into that space, and it feels a force. The changed space is called a field: a charge makes an electric field around it, and a mass makes a gravitational field around it. The field is there whether or not anything is there to feel it.

Gravitational field strength

You already know one field well. Hang a 1 kg mass from a spring balance anywhere in Bhaktapur, and it shows about 9.8 N; hang 2 kg, and it shows 19.6 N. The force on each kilogram is the same, 9.8 N. That force per kilogram is the gravitational field strength, g:

$g = \frac{F}{m}$

measured in newtons per kilogram (N/kg). It is the g you met in Grade 10 as the acceleration due to gravity, 9.8 m/s² Grade 10: Calculating g: 1 N/kg is the same as 1 m/s², since 1 N = 1 kg m/s². You also found there that $g = \dfrac{GM}{R^2}$ at the Earth’s surface, and that it gets smaller higher up. That is the strength of the Earth’s gravitational field at a distance R from its center.

Electric field strength

The electric field is described the same way. Put a small positive test charge q at a point, and measure the force F on it. The electric field strength there is the force on each coulomb:

$E = \frac{F}{q}$

measured in newtons per coulomb (N/C). The field’s direction is the direction of the force on a positive charge. A negative charge in the same field is pushed the opposite way.

For a single charge Q, Coulomb’s law gives the field at a distance r: the force on the test charge is kQq/r², so

$E = \frac{kQ}{r^2}$

just as $g = \dfrac{GM}{r^2}$ for a planet of mass M. Once you know the field at a point, the force on any charge put there is

$F = qE$

and the force on any mass is F = mg. A field is a short way of saying, for every point at once, what force a charge or a mass would feel there.

Field lines

Faraday drew fields as field lines. Draw a line that, at every point, goes the way a small positive charge (or, for gravity, a small mass) would be pushed there. The rules:

The arrows show the direction of the field: out of positive charges, into negative charges. Gravitational field lines always point into a mass, because gravity only pulls.
Field lines start on positive charges and end on negative charges (or go off to far away). They never cross: at each point the field has only one direction.
Where the lines are close together, the field is strong; where they spread apart, it is weak.

Figure 8.3 Field lines, worked out from the 1/r² law. Top left: the Earth’s gravitational field points straight in, towards its center. Top right: around a charge on its own, the electric field points straight out from a positive charge and straight in to a negative one. Bottom: with two charges, their fields add together. Lines run from a positive charge to a nearby negative one; between two positive charges the lines bend away, and at the midpoint (the small red ring) the two fields cancel exactly.

Drag the yellow test charge around. The red arrow is the force it would feel, in the field’s direction, longer where the field is stronger. Then add a second charge, negative or positive, and see how the field changes. Where is the field strongest? Can you find a place with no field at all?

Charges:

Field strength at the test charge: 2,096 N/C
Force on a +1 nC test charge: 2.1 µN

The lines start on the + charge and end on the − charge. A + test charge is pushed along them; a − one would be pulled the other way.

Worked example

The field of a proton

What is the electric field strength 5.29 × 10⁻¹¹ m from a proton (charge 1.60 × 10⁻¹⁹ C), where the electron of a hydrogen atom is? Use it to find the force on the electron, and check that you get the answer from the hydrogen atom example.

Answer

$E = \frac{kQ}{r^2} = \frac{8.99 \times 10^{9} \times 1.60 \times 10^{-19}}{(5.29 \times 10^{-11})^2} = \frac{1.44 \times 10^{-9}}{2.80 \times 10^{-21}} = 5.14 \times 10^{11}\ \text{N/C}$
pointing away from the proton. The force on the electron is
$F = qE = 1.60 \times 10^{-19}\ \text{C} \times 5.14 \times 10^{11}\ \text{N/C} = 8.22 \times 10^{-8}\ \text{N}$
the same as before. The electron is negative, so the force points the other way from the field: towards the proton, an attraction.

Try it

The Moon’s field at the Earth

The Moon (mass 7.35 × 10²² kg) is 3.84 × 10⁸ m from the Earth. a. What is the strength of the Moon’s gravitational field at the Earth? b. What force does it put on a 60 kg student? c. Compare this with the Earth’s field at the Moon (mass of the Earth 5.97 × 10²⁴ kg).

Hint

g = GM/r², then F = mg.

Answer

a. $g = \dfrac{GM}{r^2} = \dfrac{6.67 \times 10^{-11} \times 7.35 \times 10^{22}}{(3.84 \times 10^{8})^2} = \dfrac{4.90 \times 10^{12}}{1.47 \times 10^{17}}$ = 3.3 × 10⁻⁵ N/kg.
b.F = mg = 60 × 3.3 × 10⁻⁵ = 2.0 × 10⁻³ N, about the weight of a fifth of a gram. Not something you notice, though it is this field, pulling a little harder on the near side of the Earth than the far side, that raises the tides.
c. $g = \dfrac{6.67 \times 10^{-11} \times 5.97 \times 10^{24}}{1.47 \times 10^{17}}$ = 2.7 × 10⁻³ N/kg, 81 times stronger, because the Earth is 81 times as massive and the distance is the same.

Did you know?

The field that makes lightning

Dry air doesn’t normally conduct electricity. But in a field of about 3 × 10⁶ N/C, it breaks down: the field speeds up the few free electrons in the air so much that they knock electrons off the air molecules they hit, those electrons knock off more, and the air becomes a conducting path, a spark. When you touch a door handle after walking across a carpet, the spark is a few millimeters of air breaking down. In a thundercloud, collisions between ice crystals and soft hail separate huge charges, the field grows until the air breaks down, and the spark is a lightning bolt kilometers long. Lightning is a real danger in Nepal: by the count of the National Disaster Risk Reduction and Management Authority, it killed about 1,070 people between April 2014 and March 2026, nearly 90 a year. A car with a metal body is a safe place in a storm, not because of its rubber tyres (lightning has already crossed kilometers of air), but because the current runs through the metal body, round the outside of the people inside, and on to the ground.

At a certain point, the electric field points due east. An electron is placed there. Which way is the force on it?

Pick an answer to see why.

In Figure 8.3, the field lines around a single positive charge are closest together near the charge. What does that tell you?

Pick an answer to see why.

Why the square?

Both laws have an r² underneath. That isn’t a coincidence, and the field lines show why.

Imagine the field lines leaving a charge (or going into a mass) evenly in every direction, like light from a bulb. At a distance r, all of them pass through the surface of a sphere of radius r, whose area is 4πr². At twice the distance, the same lines are shared over a sphere with four times the area, so each square meter gets a quarter as many lines: the field is a quarter as strong. At three times the distance, nine times the area, a ninth of the field.

Figure 8.4 The same lines spread from a point over a square at distance r, over four such squares at 2r and over nine at 3r. Each square gets ¼ of the lines at 2r and ⅑ at 3r: the inverse square law.

So any field that spreads out evenly from a point in our three-dimensional space, and isn’t used up on the way, must weaken as 1/r². The light from a bulb and the sound from a firecracker spread the same way, which is why a lamp 2 m away gives a quarter of the light it gives at 1 m.

Think about it

Inside a hollow sphere

Cavendish’s proof of the inverse square law used a hollow charged sphere: the charge stays on its outside, and there is no field inside. Newton had shown the same for gravity: inside a hollow shell of mass, the pulls of all its parts cancel. Why might the near part of the shell, pulling one way, cancel the much bigger far part, pulling the other?

Answer

From a point inside the shell, look out in two opposite directions through a narrow cone. The cone meets a small patch of the shell on the near side and a bigger patch on the far side. The far patch’s area is bigger in proportion to its distance squared, so it holds more charge (or mass) in exactly that proportion; but its pull is weaker by exactly the same factor, 1/r². The two cancel, for every pair of cones, so the total force is zero. That only works if the force goes as exactly 1/r²: with any other power, the canceling fails and there would be a force inside. So finding no force inside is a test of the law.

Two forces on one drop

Both laws can act on the same thing at once. The most famous example is an experiment that measured the charge of a single electron.

Figure 8.5 Millikan’s oil-drop experiment. A fine spray of oil falls through a hole into the space between two metal plates, joined to a battery so that the top plate is positive. The field E points down. A drop that has picked up extra electrons feels an electric force qE upwards, against its weight mg. The drop is far smaller than drawn.

A tiny drop of oil between the plates is pulled down by gravity with its weight, mg. If it has picked up a few spare electrons, the field pushes it up with a force qE. Turn up the voltage, and the field gets stronger, until the two forces are equal and the drop floats:

$qE = mg \qquad\text{so}\qquad q = \frac{mg}{E}$

Measure the drop’s mass and the field that holds it still, and you have its charge.

An open metal drum with a tap, a side tube and a small round window, its heavy bolted lid hanging above it, in a museum case — **Figure 8.6** Part of an oil-drop apparatus from Millikan’s laboratory, made about 1916, now in the Museum of Science and Industry in Chicago. Photo: Daderot, CC0

In short

There are two kinds of charge. Like charges repel; unlike charges attract. Rubbing moves electrons; charge is never made or destroyed.
Coulomb’s law: $F = \dfrac{k q_1 q_2}{r^2}$ , with k = 8.99 × 10⁹ N m²/C². The same shape as Newton’s law of gravitation, but it can push as well as pull.
Between two protons the electric force is about 10³⁶ times their gravity, at any distance. Gravity rules on large scales only because masses add up, while positive and negative charges cancel.
A field gives the force per unit at each point: gravitational field strength g = F/m (N/kg), electric field strength E = F/q (N/C). Near a point mass or charge, g = GM/r² and E = kQ/r².
Field lines go out of positive charges and into negative charges and masses; they are close together where the field is strong.
Both forces fall off as 1/r², because the field spreads over a sphere of area 4πr².

Questions

Two small charged spheres attract each other with a force of 6.0 × 10⁻³ N when they are 4.0 cm apart. What is the force when they are 12 cm apart?

Pick an answer to see why.

Which is true of both the gravitational force and the electric force between two small objects?

Pick an answer to see why.

a.

Two identical metal spheres carry charges of +6.0 µC and −2.0 µC, with their centers 30 cm apart.

i.Calculate the force between them, and say whether it is attractive or repulsive.
ii.The spheres are touched together, so that their charges share out equally, and then put back 30 cm apart. What is the charge on each now, and what is the new force?

Answer

i. $F = \dfrac{8.99 \times 10^{9} \times (6.0 \times 10^{-6}) \times (-2.0 \times 10^{-6})}{(0.30)^2} = \dfrac{-0.108}{0.090} = -1.2\ \text{N}$ . A force of 1.2 N, attractive (opposite charges).
ii.The total charge is conserved: +6.0 − 2.0 = +4.0 µC, shared equally, so each sphere has +2.0 µC. Now $F = \dfrac{8.99 \times 10^{9} \times (2.0 \times 10^{-6})^2}{0.090} = 0.40\ \text{N}$ , repulsive, because both are now positive.

b.

The Earth (mass 5.97 × 10²⁴ kg) and the Moon (mass 7.35 × 10²² kg) are 3.84 × 10⁸ m apart.

i.Calculate the gravitational force between them.
ii.Suppose the Earth and the Moon each carried an equal positive charge Q. How big would Q have to be for the electric push to cancel gravity’s pull?
iii.Using your answer, explain why astronomers can ignore electric forces when they work out the Moon’s orbit.

Answer

i. $F = \dfrac{6.67 \times 10^{-11} \times 5.97 \times 10^{24} \times 7.35 \times 10^{22}}{(3.84 \times 10^{8})^2} = \dfrac{2.93 \times 10^{37}}{1.47 \times 10^{17}} = 1.99 \times 10^{20}\ \text{N}$ , about 2.0 × 10²⁰ N.
ii.The electric push $kQ^2/r^2$ must equal F, so $Q^2 = \dfrac{F r^2}{k} = \dfrac{2.93 \times 10^{37}}{8.99 \times 10^{9}} = 3.26 \times 10^{27}$ , and Q = 5.7 × 10¹³ C.
iii.That is a huge charge, about 3.6 × 10³² spare electrons each. A body with that much spare charge would pull in charge of the opposite sign from the gas in space around it until it was neutral again; real planets carry nothing like it. So the electric force between them is negligible, and gravity alone decides the orbit. (The Moon has about 10⁴⁹ electrons, so this would be only about one spare electron in 10¹⁷, yet still far more than it has.)

c.

Explain, using the idea of charge, why a whole planet hardly pulls or pushes a charged balloon with its electric force, but a comb charged by a few strokes through your hair can lift scraps of paper against the gravity of the whole Earth.

Answer

A planet contains enormous numbers of positive and negative charges, but almost exactly equal numbers of each. The pull of its positive charges on the balloon is canceled by the push of its negative charges, so the total electric force is almost zero. Its mass, on the other hand, is all of one kind, and every kilogram adds its pull, so the planet’s gravity is large. The comb carries only a tiny charge, but that charge is unbalanced, and the electric force is so much stronger than gravity (about 10³⁶ times, between two protons) that even a tiny unbalanced charge a few centimeters away can beat the gravity of the whole Earth on a scrap of paper weighing a few milligrams.

d.

A small positive charge is placed at a point P, 3.0 cm from a charge of +2.0 nC.

i.What is the electric field strength at P, and which way does it point?
ii.A charge of −5.0 nC replaces the small charge at P. What force does it feel, and which way?
iii.Without calculating, what would the field strength be at 6.0 cm from the +2.0 nC charge?

Answer

i. $E = \dfrac{kQ}{r^2} = \dfrac{8.99 \times 10^{9} \times 2.0 \times 10^{-9}}{(0.030)^2} = \dfrac{17.98}{9.0 \times 10^{-4}} = 2.0 \times 10^{4}\ \text{N/C}$ , pointing away from the +2.0 nC charge.
ii.F = qE = 5.0 × 10⁻⁹ × 2.0 × 10⁴ = 1.0 × 10⁻⁴ N, towards the +2.0 nC charge, because the charge is negative.
iii.Twice the distance, so a quarter of the field: 5.0 × 10³ N/C.

e.

The table shows the force measured between two charged balls at different distances, with the charges kept the same.

Distance between centers (cm)	2.0	3.0	4.0	6.0
Force (mN)	36	16	9.0	4.0

i.Show that the results agree with Coulomb’s law.
ii.Predict the force at 8.0 cm.
iii.Each ball carries the same charge. Use the result at 2.0 cm to find it.

Hint

If F ∝ 1/r², then F × r² is the same for every row.

Answer

i.Work out F × r² for each: 36 × 2.0² = 144; 16 × 3.0² = 144; 9.0 × 4.0² = 144; 4.0 × 6.0² = 144 (mN cm²). The product is constant, so the force is inversely proportional to the square of the distance, as Coulomb’s law says.
ii.F = 144 ÷ 8.0² = 2.25 mN, about 2.3 mN.
iii.F = kq²/r², so $q^2 = \dfrac{F r^2}{k} = \dfrac{0.036 \times (0.020)^2}{8.99 \times 10^{9}} = 1.6 \times 10^{-15}$ , and q = 4.0 × 10⁻⁸ C (40 nC).

f.

In Millikan’s experiment, a drop of mass 4.9 × 10⁻¹⁵ kg is held still between the plates by a field of 1.0 × 10⁵ N/C.

i.Draw the forces on the drop and say what is true of them.
ii.Find the charge on the drop, and how many electrons it is.
iii.If the field were switched off, what would happen to the drop, and why?

Answer

i.Two forces: its weight, mg, down, and the electric force, qE, up. The drop is still, so they are equal and opposite.
ii.q = mg/E = 4.9 × 10⁻¹⁵ × 9.8 ÷ 1.0 × 10⁵ = 4.8 × 10⁻¹⁹ C, which is 4.8 × 10⁻¹⁹ ÷ 1.60 × 10⁻¹⁹ = 3 electrons.
iii.Only its weight would act, so it would fall. (It would quickly reach a slow, steady speed, because air resistance on such a tiny drop is large compared with its weight; Millikan used that speed to find each drop’s mass.)

Where to go next

Why does salt melt at 801 °C but ice at 0 °C? The electric force between ions and molecules decides how hard they are to pull apart.
Why is spider silk so strong? The strength of every material comes from electric forces between its atoms and molecules, and how they are arranged.
How do magnets push without touching? Magnetic fields, the energy stored in fields, and how electric currents make them.
Why do airbags save lives? Momentum and impulse: why spreading a collision over a longer time makes the force smaller.