How Do Magnets Push Without Touching?: Going Further 9–12

Two magnets, one field

You met magnets in Grade 8: every magnet has a north and a south pole, unlike poles attract and like poles repel, and the region round a magnet where its pull can be felt is its magnetic field Grade 8: Magnet. In Grade 10 you found that an electric current makes a magnetic field too, that a coil carrying a current behaves like a bar magnet, and that a field can be drawn as field lines, crowded together where it is strong (The magnetic field around a solenoid, Magnetic flux). And in Why Is Gravity So Weak? you met electric fields: a field says, for every point at once, what force something would feel if it were put there.

So far you have drawn the field of one magnet. But a pull needs two things. Put a second magnet near the first, and each one sits in the other’s field. The field you would map with a compass is now the two fields added together, point by point, and its shape is quite different from either magnet’s field alone.

Two bar magnets, drawn to scale, with the field lines of both together. Turn one magnet round, and move them closer and farther apart. When do lines cross the gap? Where do they crowd together?

Facing poles:

Gap between the magnets: 20 mm

North faces south: lines from the N pole run straight across the gap into the S pole. Like stretched rubber bands, they pull the magnets together. Close the gap and they crowd closer: a stronger pull.

The English scientist Michael Faraday found a way to read the force straight from a picture like this. Imagine that every field line is a stretched rubber band: it always tries to get shorter, and it pushes sideways on the lines beside it, so lines squashed together try to spread out.

Unlike poles facing. Lines leave the N pole and run straight across the gap into the S pole. Like stretched bands, they try to shorten, and they pull the magnets together. The closer the magnets, the more lines crowd into the gap, and the stronger the pull.
Like poles facing. No line crosses the gap. Lines from both N poles crowd into the space between them, press sideways against each other, and turn away. Their sideways push pushes the magnets apart. Halfway between two equal magnets, the two fields are equal and opposite and cancel: a point with no field at all.

This picture isn’t just a memory aid. Physicists later showed that the forces worked out this way, from the tension along the lines and their sideways pressure, are exactly the forces the magnets really feel.

Iron filings on a white sheet over two bar magnets placed end to end with a gap between them; the filings form lines that run across the gap from one magnet to the other, and curves looping round the outside — **Figure 12.1** Iron filings over two bar magnets, south pole facing north. The filings line up along the field: across the gap they form lines from one magnet to the other. Photo: MikeRun , Vera Wurmsdobler, CC BY-SA 4.0

Iron filings on a white sheet over two bar magnets placed end to end, like poles facing; the filings between the magnets bend away sideways, and none runs from one magnet to the other — **Figure 12.2** The same magnets with like poles facing. No lines cross the gap: they turn aside. At the very center of the gap, where the two fields cancel, the filings have no direction to line up in. Photo: MikeRun , Vera Wurmsdobler, CC BY-SA 4.0

Magnetic field lines never end

There is one big difference between magnetic and electric field lines. Electric field lines start on positive charges and end on negative ones (Field lines). Magnetic field lines have no start or end. A line leaves the N pole, curves round outside, enters the S pole, and carries on through the inside of the magnet back to the N pole: every magnetic field line is a closed loop.

That is why you can never cut a magnet into a north half and a south half. Every piece has both poles Grade 8: Molecular theory of magnetism. A lone north pole, a magnetic monopole, would be a place where field lines start, like a positive charge. The physicist Paul Dirac showed in 1931 that the laws of physics allow one, and physicists have searched for them ever since, in rocks, in cosmic rays and at particle colliders. None has been found (as of 2026).

Activity

Make your own iron filings

You need: a strong magnet (a fridge magnet is too weak; the ring from an old loudspeaker or a small neodymium disc is good), two thin plastic bags, a cup of dry sand from a riverbank or building site, a sheet of white card, and a second magnet.

Put the magnet inside one plastic bag, and stir it through the dry sand for a minute.
Hold the bag over a sheet of paper and pull the magnet out of the bag. The black grains that clung to the bag fall onto the paper. These are grains of magnetite, an iron ore that a magnet attracts; most sand has some.
Put both magnets in the second bag (so the grains can’t stick to them), a few centimeters apart, north facing south. Lay the card on top.
Sprinkle the black grains thinly over the card and tap it gently. Look at the pattern in the gap.
Turn one magnet round, so like poles face, and repeat.

Strong magnets snap together hard enough to pinch skin and chip each other. Keep them away from small children: a child who swallows two magnets can be badly hurt as they pull together inside.

What you should see

The grains line up in curves from one magnet to the other, crowded and straight across the gap, like the photos above.
The lines in the gap bend away to the sides, and the middle of the gap is nearly empty: there, the two fields cancel.

A story from science

The bookbinder who drew the invisible

Michael Faraday (1791–1867) was the son of a poor blacksmith in London and left school at 13. At 14 he was apprenticed to a bookbinder, and he read the science books he was binding. In 1812 a customer gave him tickets to the lectures of the famous chemist Humphry Davy; Faraday wrote up his notes, bound them, and sent them to Davy, and a few months later became his laboratory assistant. Faraday never learned much mathematics. Instead he thought in pictures. Where others imagined magnets pulling on each other directly across empty space, Faraday pictured the space itself filled with lines of force, and spent years mapping them with compasses and iron filings, fixing the filings on waxed paper to keep them. In 1855 the young Scottish physicist James Clerk Maxwell turned Faraday’s lines into mathematics, and ten years later his equations showed that a ripple in electric and magnetic fields would travel at the speed of light. Light itself is such a ripple: the fields Faraday drew can carry energy across the universe.

Two bar magnets lie end to end with a gap between them. In the gap, the field lines bend away to the sides and none crosses from one magnet to the other. What can you say?

Pick an answer to see why.

Why can’t you cut a bar magnet to get a piece with only a north pole?

Pick an answer to see why.

Pulling apart stores energy

To pull the magnets apart, you must keep pulling against their attraction while they move. In Where Does the Energy Go When a Ball Bounces? you met work: when a force moves something, work = force × distance, and work is energy passed from one thing to another. So pulling magnets apart takes energy out of your muscles.

That energy isn’t lost. Let go, and the magnets’ pull speeds the magnet up: it gains kinetic energy. Then they hit, and the kinetic energy becomes sound and heat. While you held them apart, the energy was stored, like the energy of a ball held up high. The ball’s energy belongs to the ball and the Earth together, stored in the gravitational field between them (Gravitational potential energy). In the same way, the magnets’ energy belongs to the pair, and it is stored in the magnetic field between them. A single magnet alone on a table has no energy of this kind to give.

Like poles work the other way round. Pushing them together takes work, like squashing a spring, and letting go gives it back as the magnet flies off. Either way, the rule is the same: when two objects that push or pull each other through a field are moved against the force, energy is stored in the field; when they move the way the force pushes them, the field gives energy back.

When the force changes as you pull

Lifting a ball is easy to count, because its weight is the same all the way up: work = mg × h. A magnet’s pull is not like that. It is enormous when the magnets touch, and falls away to almost nothing a few centimeters apart.

The table gives the force between two neodymium disc magnets, 10 mm across and 5 mm thick, a common size (worked out from the strength of their material, not measured):

Gap between the magnets (mm)	0	1	2	3	4	6	8	10
Force (N)	28.3	16.2	10.9	7.75	5.65	3.17	1.89	1.18

Pulling them apart by a millimeter near contact takes far more work than a millimeter farther out. To add it up, split the pull into small steps. Over each step the force hardly changes, so the work is about (the average force) × (the step), the area of a thin strip under the force–distance graph. Add up the strips, and you have the whole area. The work done is the area under the force–distance graph: for a constant force, the area is just a rectangle, force × distance.

Worked example

How much energy is in the gap?

Use the table to estimate the work needed to pull the two magnets from touching to 10 mm apart. If all of that energy became kinetic energy of one magnet (mass 2.95 g) as it snapped back, the other being held still, how fast would it be moving when they hit?

Answer

Take each step in the table as a strip with a straight top (a trapezium), area = average force × width. With the force in newtons and the width in millimeters, each area is in N mm, which is millijoules (mJ), since 1 N × 1 mm = 0.001 N m = 1 mJ.

Step (mm)	Average force (N)	Width (mm)	Work (mJ)
0 to 1	(28.3 + 16.2) ÷ 2 = 22.25	1	22.25
1 to 2	(16.2 + 10.9) ÷ 2 = 13.55	1	13.55
2 to 3	(10.9 + 7.75) ÷ 2 = 9.33	1	9.33
3 to 4	(7.75 + 5.65) ÷ 2 = 6.70	1	6.70
4 to 6	(5.65 + 3.17) ÷ 2 = 4.41	2	8.82
6 to 8	(3.17 + 1.89) ÷ 2 = 2.53	2	5.06
8 to 10	(1.89 + 1.18) ÷ 2 = 1.54	2	3.07
total			about 69 mJ

So it takes about 69 mJ (0.069 J) to pull them 10 mm apart, and that much is stored in the field. (With far more, thinner strips, the area comes to 67 mJ. Straight-topped strips over-count a curve that sags like this one.)
Speed: all of it becomes kinetic energy, ½mv² = 0.067 J (Kinetic energy), with m = 0.00295 kg:
$v = \sqrt{\frac{2 \times 0.067\ \text{J}}{0.00295\ \text{kg}}} = \sqrt{45\ \text{m}^2/\text{s}^2} = 6.7\ \text{m/s}$
About 6.7 m/s, 24 km/h, about as fast as you can sprint, built up over just 10 mm. No wonder strong magnets chip when they snap together.

The same two magnets, the left one held still. Set the gap, press Let go, and watch the shaded area and the energy bars. Then try like poles, pushed close together.

Facing poles:

Starting gap: 10 mm

Force at the start: 1.2 N
Top speed: 6.7 m/s

You have pulled the magnets 10 mm apart. The work you did is stored in the field: the shaded area under the force curve. Press Let go.

You push two magnets together, north pole to north pole, hold them 2 mm apart, and let go. What happens to the energy?

Pick an answer to see why.

The energy of two charges

Charges do exactly what magnets do. Two unlike charges pull each other; pull them apart, and you store energy in the electric field between them. Two like charges push; push them together, and you store energy. Because Coulomb’s law gives the force exactly (Coulomb’s law), the stored energy can be worked out exactly too.

The force between two small charges is $F = kq_1q_2/r^2$ . Add up the strips under this curve from a distance r all the way out to “far apart”, and mathematics gives a simple answer. The electric potential energy of two charges a distance r apart is

$U = \frac{k\,q_1 q_2}{r}$

counting zero when they are far apart. U is in joules, q₁ and q₂ in coulombs, r in meters, and k = 8.99 × 10⁹ N m²/C² as before. Notice that it is r, not r²: double the distance and the stored energy halves, while the force falls to a quarter.

Like charges: q₁q₂ is positive, so U is positive. It is the energy the pair will give out if they are let go and fly apart.
Unlike charges: q₁q₂ is negative, so U is negative. Negative only means less than when far apart: coming together from far away, they have already given out that much energy, and you must put it back to pull them apart again.

Figure 12.3 The energy stored by two charges of 50 nC, against the distance between them, from

U = kq_1q_2/r

. Like charges (red) store less energy as they fly apart; unlike charges (blue) store more as they are pulled apart, rising towards zero, the energy they have when far apart.

Worked example

Two balls fly apart

Two small balls each carry +50 nC. One is fixed; the other, of mass 1.0 g, can move freely without friction. They start 3.0 cm apart and are let go. a. How much energy is stored at the start? b. How fast is the free ball moving when they are 30 cm apart?

Answer

a. $U = \dfrac{kq_1q_2}{r} = \dfrac{8.99 \times 10^{9} \times (50 \times 10^{-9})^2}{0.030} = \dfrac{2.25 \times 10^{-5}}{0.030} = 7.5 \times 10^{-4}\ \text{J}$
b.At 30 cm: $U = \dfrac{2.25 \times 10^{-5}}{0.30} = 7.5 \times 10^{-5}\ \text{J}$ .

The field has given out 7.5 × 10⁻⁴ − 0.75 × 10⁻⁴ = 6.7 × 10⁻⁴ J, and it has all become kinetic energy:
$v = \sqrt{\frac{2 \times 6.7 \times 10^{-4}\ \text{J}}{0.0010\ \text{kg}}} = 1.2\ \text{m/s}$
The ball reaches about 1.2 m/s. Moving on from 30 cm to far away would add only a little more: 90 % of the stored energy has already been given out.

Try it

Pulling unlike charges together

Charges of +40 nC and −40 nC are 20 cm apart. How much energy does the field give out as they move to 5.0 cm apart?

Hint

Work out U at both distances; the energy given out is the difference.

Answer

$kq_1q_2 = 8.99 \times 10^{9} \times (40 \times 10^{-9}) \times (-40 \times 10^{-9}) = -1.44 \times 10^{-5}\ \text{J m}$ .
At 20 cm: U = −1.44 × 10⁻⁵ ÷ 0.20 = −7.2 × 10⁻⁵ J. At 5.0 cm: U = −1.44 × 10⁻⁵ ÷ 0.050 = −2.88 × 10⁻⁴ J.
The stored energy has fallen by 2.88 × 10⁻⁴ − 0.72 × 10⁻⁴ = 2.2 × 10⁻⁴ J, which the field gives out (as kinetic energy, if nothing stops them).

Worked example

What holds a hydrogen atom together?

In a hydrogen atom, the electron is about 5.29 × 10⁻¹¹ m from the proton (Inside a hydrogen atom). a. What is their electric potential energy? b. For an electron going round in a circle, the pull of the proton keeps it moving at just the speed at which its kinetic energy is half the size of $ke^2/r$ (we won’t prove that here). What is its kinetic energy? c. How much energy must be given to the atom to pull the electron away completely?

Answer

a. $U = \dfrac{k q_1 q_2}{r} = \dfrac{8.99 \times 10^{9} \times (1.60 \times 10^{-19}) \times (-1.60 \times 10^{-19})}{5.29 \times 10^{-11}} = -4.35 \times 10^{-18}\ \text{J}$
b.Kinetic energy = ½ × 4.35 × 10⁻¹⁸ = 2.18 × 10⁻¹⁸ J.
c.The total energy of the atom is kinetic + potential = 2.18 × 10⁻¹⁸ − 4.35 × 10⁻¹⁸ = −2.18 × 10⁻¹⁸ J. Far apart and at rest, the electron and proton would have zero of both. So to pull the atom apart you must give it 2.18 × 10⁻¹⁸ J.

This is just the energy measured in experiments to pull the electron off a hydrogen atom (called 13.6 electronvolts). An atom’s energy, like a ball’s, is energy of motion plus energy of position in a field.

Two like charges are 10 cm apart. They are moved to 20 cm apart. What happens to the electric potential energy?

Pick an answer to see why.

Energy between charged plates

Take two flat metal plates, face to face a little way apart, and join them to a battery. Electrons flow off one plate and onto the other, until one plate is positive, the other negative, and the potential difference between them is the battery’s voltage. Two charged plates like this make a capacitor, a store for electric energy that is in almost every electronic circuit.

Between the plates, the field lines run straight across from the + plate to the − plate, parallel and evenly spaced (except near the edges): the field is uniform, the same strength and direction everywhere between the plates, like the field inside a solenoid. So a charge q between the plates feels the same force F = qE wherever it is (Electric field strength), just as a ball near the ground feels the same weight mg at any height.

So carrying a positive charge across the gap d, from the − plate to the + plate, against the field, is like lifting a ball: the force is constant, and the work is force × distance.

Figure 12.4 Two models of the same idea. Lifting a ball against the Earth’s uniform gravitational field stores mgh. Carrying a positive charge from the − plate to the + plate, against the uniform electric field between them, stores qEd. Neither is to scale.

$\text{work} = F \times d = qEd$

You learned in Grade 9 that the potential difference between two points is the energy per coulomb, V = W/Q, so moving a charge q across a potential difference V takes energy qV Grade 9: Potential difference. Both describe the same trip, so qEd = qV, and

$E = \frac{V}{d}$

The field between the plates is the voltage divided by the gap. This gives a second unit for field strength, the volt per meter (V/m), the same as the newton per coulomb: 1 V/m = 1 J/(C m) = 1 N/C.

	A ball near the ground	A charge between plates
the field	g, about 9.8 N/kg, uniform	E = V/d, uniform
the force	mg	qE
energy stored, moved against the field	mgh	qEd = qV
let go	it falls, and gains kinetic energy	it is pushed back across, and gains kinetic energy

Worked example

An electron gun

In a laboratory electron tube, electrons leave a hot metal plate and are pulled towards a second plate 2.0 cm away, with 2,000 V between the plates. a. What is the field between the plates? b. How much energy does an electron gain crossing the gap? c. How fast is it going when it arrives? (Charge of an electron 1.60 × 10⁻¹⁹ C, mass 9.11 × 10⁻³¹ kg.)

Answer

a. $E = \dfrac{V}{d} = \dfrac{2000\ \text{V}}{0.020\ \text{m}} = 1.0 \times 10^{5}\ \text{V/m}$ (or N/C).
b.Energy = qV = 1.60 × 10⁻¹⁹ C × 2000 V = 3.2 × 10⁻¹⁶ J. (Or qEd = 1.60 × 10⁻¹⁹ × 1.0 × 10⁵ × 0.020: the same.) The electron is negative, so the field pushes it towards the + plate: like a ball falling, it loses stored energy and gains kinetic energy.
c. $v = \sqrt{\dfrac{2 \times 3.2 \times 10^{-16}}{9.11 \times 10^{-31}}} = \sqrt{7.0 \times 10^{14}} = 2.7 \times 10^{7}\ \text{m/s}$

About 27,000 km every second, nearly a tenth of the speed of light, from just 2 cm of field. (At much higher voltages the speed gets near the speed of light, and this simple formula for kinetic energy no longer works.)

How much energy is stored in a capacitor?

Charging the plates means carrying charge from one plate to the other (the battery does the carrying). The first bit of charge crosses when the plates are still uncharged, with no voltage between them, so it takes almost no work. But each bit carried across makes the voltage a little higher: the voltage rises in step with the charge. The last bit is carried across the full voltage V. On average, each coulomb is carried across half the final voltage, so a capacitor holding charge Q at voltage V stores

$\text{energy stored} = \tfrac{1}{2}\,QV$

On a graph of voltage against charge, it is the area of a triangle.

Figure 12.5 Charging a capacitor. The voltage between the plates rises in step with the charge carried across. Each small charge is carried across the voltage of its moment (the strips), and together they fill the triangle: energy = ½QV.

Think about it

Pulling the plates apart

A capacitor is charged and then disconnected from the battery, so its charge can’t change. The two plates attract each other (one is +, the other −). You pull them farther apart. The field between them stays the same strength, because it depends only on how much charge there is on each square meter of the plates. What happens to the voltage and to the stored energy? Where does the extra energy come from, and where is it?

Answer

V = Ed: the field is the same and the gap is bigger, so the voltage rises. The stored energy, ½QV, rises with it. The extra energy is the work you did pulling the plates apart against their attraction. And it is stored in the extra space filled with field: the field is the same everywhere between the plates, so each cubic meter of it holds the same energy, and a wider gap holds more cubic meters of field. This is the clearest sign that the energy is really in the field, in the space between the plates, not on the plates themselves.

Where the energy is: in the field

The capacitor shows it most clearly: energy is stored in the space where the field is, and the stronger the field, the more energy each cubic meter holds. (The energy in each cubic meter goes as the field strength squared: double the field, four times the energy.) The same is true of magnetic fields. When two magnets snap together, the strong field that filled the gap between them vanishes, and its energy appears as the magnet’s motion. And light, which is a traveling ripple in electric and magnetic fields, carries energy across space from the Sun with no matter at all to carry it.

Did you know?

Why motors use magnets

How much energy can a field hold? An electric field in air can’t be made stronger than about 3 × 10⁶ V/m, or the air breaks down and sparks (The field that makes lightning). At that field, each cubic meter of air holds only about 40 J. A magnetic field of 1 tesla (the unit of magnetic field; a neodymium magnet’s surface is about half a tesla) holds about 400,000 J in each cubic meter, ten thousand times as much, and air doesn’t break down. That is why motors, generators, transformers and induction cookers all work with magnetic fields, not electric ones: a magnetic field can pack far more energy into a small space. Even so, fields are poor fuel tanks: a cubic meter of petrol holds about 34,000,000,000 J of chemical energy (which is itself energy stored in the electric fields between atoms).

Motion or position, at every scale

In Every kind of energy is motion or position you saw that every kind of energy is energy of motion (kinetic) or energy of position (potential). Now you can say where the energy of position is: in a field, between things that push or pull each other.

Example	Energy of motion	Energy of position: in which field?
a ball held up, then falling	the falling ball	gravitational field of the ball and the Earth
two magnets snapping together	the moving magnet	magnetic field between the magnets
a charged camera-flash capacitor	none, until it flashes	electric field between the plates
a hydrogen atom	the electron going round	electric field of the proton and the electron
a cup of hot tea	the jiggling molecules	electric fields between neighboring molecules
a stopped magnet, just after the click	faster jiggling of the molecules in the magnets and the air (sound, then heat)	the stretched and squeezed bonds between those molecules

At every scale, from an atom to a planet, energy is accounted for in just these two ways. When it seems to disappear, as after the click, it has gone into the motion and positions of molecules, too small and too many to see.

A student says: “Magnets are a free source of energy. They snap together again and again, and never run down.” What is wrong with this?

Pick an answer to see why.

A charged capacitor is disconnected and left on a table. Where is its stored energy?

Pick an answer to see why.

In short

Two magnets, or two charges, each sit in the other’s field. The field lines of the two add together: between unlike poles they run across the gap and pull like stretched bands; between like poles they crowd, turn aside and push, with a point of no field between equal magnets.
Magnetic field lines are closed loops: they never start or end, so there is no lone magnetic pole (none found, as of 2026). Electric field lines start on + charges and end on − charges.
Moving two objects against the force between them (pulling unlike poles or charges apart, pushing like ones together) stores energy in the field. Let go, and the field gives it back as kinetic energy.
Work = area under the force–distance graph, whether the force is constant or not.
Two charges a distance r apart: $U = \dfrac{kq_1q_2}{r}$ , zero when far apart, negative for unlike charges.
Between charged plates the field is uniform, $E = \dfrac{V}{d}$ , and moving a charge across stores qEd = qV, as lifting stores mgh. A capacitor charged to V with charge Q stores ½QV.
Energy is stored in the field itself, in the space where it is. All energy is energy of motion or energy of position in a field.

Questions

Two unlike charges are let go from rest and move towards each other. Which describes the energy of the pair?

Pick an answer to see why.

The voltage between two parallel plates is doubled, and the gap between them is also doubled. The field between them

Pick an answer to see why.

a.

A ring magnet floats above another ring magnet on a pencil, like poles facing, with a gap of 8 mm. You press the top magnet down until the gap is 4 mm, and let go. It shoots up, rises above 8 mm, falls back, and bobs up and down a few times before it settles at 8 mm again.

i.Why does the top magnet float at 8 mm?
ii.Draw an energy model: three bar charts showing the gravitational, magnetic and kinetic energy of the system, and the thermal energy, when you hold the magnet at 4 mm, when it passes 8 mm on the way up, and at the top of its first bounce.
iii.Why does it stop bobbing?

Answer

i.At 8 mm, the magnetic push up on the top magnet is exactly equal to its weight. Closer, the push is stronger than the weight; farther, it is weaker. So 8 mm is where the forces balance.
ii.Holding at 4 mm: a lot of magnetic energy (you pushed like poles closer, against the push), the least gravitational energy (the magnet is at its lowest), no kinetic energy, no extra thermal energy. Passing 8 mm going up: magnetic energy much smaller (the poles are farther apart), gravitational energy a little bigger (it is higher), kinetic energy large (it is moving fastest near here, where the forces balance), a little thermal energy (friction on the pencil, air). Top of the first bounce: magnetic energy smallest, gravitational energy largest, kinetic energy zero (it stops for an instant), a little more thermal energy. The total of all the bars is the same each time, and more than when the magnet floated at rest by exactly the work you did pressing it down.
iii.Each bob, friction against the pencil and the air turns some of the energy into thermal energy (molecular motion) and sound. When all of the extra energy you put in has gone that way, the magnet is left at rest where the forces balance.

b.

Two small spheres, each with a charge of +30 nC, are held 5.0 cm apart. One is fixed; the other has a mass of 0.50 g and is free to move without friction.

i.Calculate the force between them, and the electric potential energy.
ii.The free sphere is let go. How fast is it moving when the spheres are 15 cm apart?
iii.Explain why it speeds up quickly at first and then more and more slowly.

Answer

i. $F = \dfrac{8.99 \times 10^{9} \times (30 \times 10^{-9})^2}{(0.050)^2} = \dfrac{8.09 \times 10^{-6}}{2.5 \times 10^{-3}} = 3.2 \times 10^{-3}\ \text{N}$ , a push. $U = \dfrac{8.09 \times 10^{-6}}{0.050} = 1.62 \times 10^{-4}\ \text{J}$ .
ii.At 15 cm: U = 8.09 × 10⁻⁶ ÷ 0.15 = 0.54 × 10⁻⁴ J. Energy given out = 1.62 × 10⁻⁴ − 0.54 × 10⁻⁴ = 1.08 × 10⁻⁴ J = ½mv², so $v = \sqrt{\dfrac{2 \times 1.08 \times 10^{-4}}{5.0 \times 10^{-4}}} = 0.66\ \text{m/s}$ .
iii.The push is $kq_1q_2/r^2$ : strong when the spheres are close, and falling to a ninth by the time they are three times as far apart. So the acceleration is large at first and gets smaller and smaller. The sphere never stops speeding up, but it gains less and less, heading for a top speed set by all of the starting energy, 0.80 m/s.

c.

In a hospital X-ray tube, electrons are sped up from rest across 100,000 V, then hit a block of tungsten.

i.How much kinetic energy does each electron have when it reaches the tungsten?
ii.About 1 % of that energy comes out as X-rays. What happens to the rest, in terms of the motion of particles? Why must the tungsten block be cooled?

Answer

i.Energy = qV = 1.60 × 10⁻¹⁹ C × 100,000 V = 1.6 × 10⁻¹⁴ J: the electric field in the tube has given each electron that much kinetic energy.
ii.The other 99 % becomes thermal energy: each electron’s orderly motion is shared out among the atoms of the tungsten, which jiggle faster (and the jiggling is the electric fields between neighboring atoms being stretched and squeezed, back and forth). With many electrons a second, that heats the block strongly, so it must be cooled, often by spinning it so the beam keeps hitting a new spot, and with oil round the tube.

d.

A student measures the force needed to hold two magnets apart at different gaps, using a kitchen scale. (These results are made up for this question, but they are like real ones.)

Gap (mm)	5	10	15	20	30
Force (N)	4.0	2.0	1.1	0.6	0.2

i.Plot a graph of force against gap. Is the force inversely proportional to the gap? Explain.
ii.Estimate the energy stored in the field when the magnets are pulled from 5 mm to 30 mm apart.
iii.Is your estimate too big or too small? Why?

Hint

For (i), test whether force × gap is the same each time. For (ii), use strips with straight tops, and remember 1 N × 1 mm = 1 mJ.

Answer

i.Force × gap: 20, 20, 16.5, 12, 6 (N mm). It isn’t constant, so the force is not inversely proportional to the gap: it falls off faster. (For magnets far apart compared with their size, the force falls as about 1/(distance)⁴.)
ii.Strips: 5 to 10: (4.0 + 2.0) ÷ 2 × 5 = 15; 10 to 15: (2.0 + 1.1) ÷ 2 × 5 = 7.75; 15 to 20: (1.1 + 0.6) ÷ 2 × 5 = 4.25; 20 to 30: (0.6 + 0.2) ÷ 2 × 10 = 4.0. Total about 31 mJ (0.031 J).
iii.Slightly too big. The curve sags below the straight line joining each pair of points, so each straight-topped strip includes a little extra area. (Measuring at more gaps would make the strips thinner and the estimate better.)

e.

Compare the two systems: a ball and the Earth; a positive charge between two charged plates.

i.For each, say what the field is, what force acts, and what happens to the energy when the object moves with the force and against it.
ii.In what way are they alike, and in what ways different?

Answer

i.Ball and Earth: a uniform gravitational field g near the ground; the force is the weight mg, down. Lifted (against the force), it gains gravitational potential energy mgh; falling (with the force), it loses that and gains kinetic energy. Charge between plates: a uniform electric field E = V/d, from + to −; the force is qE, towards the − plate. Carried towards the + plate (against the force), it gains electric potential energy qEd = qV; moving towards the − plate (with the force), it loses that and gains kinetic energy.
ii.Alike: a uniform field, a constant force, energy stored = force × distance moved against it, stored in the field, and given back as kinetic energy. Different: gravity only ever pulls, so every mass is pulled the same way, while a negative charge between the plates is pushed the opposite way to a positive one; the electric field can be switched off or reversed with the battery; and the electric force on a small charged object can be far bigger than its weight.

Where to go next

Where does the energy go when a ball bounces? Counting energy in joules: kinetic, gravitational and thermal, and following every joule of a bouncing ball.
Why does heat never flow from cold to hot? Why the energy of the click spreads out into heat and never gathers itself back to push the magnets apart.
How much electricity can a small stream make? Stored energy at work: a stream’s drop, a turbine, and a generator turning motion into electricity.