Electric fields — what they are and how to draw them — Physics

Before this

Gravitational field strength g
Charging objects · coming soon
Electric fields — forces without touching · coming soon

This leads to

Coulomb's law
Electric field strength, E = F/q and E = V/d
Electric potential and potential energy · coming soon

A bee that feels a flower before it lands

In 2013, researchers at the University of Bristol showed that bumblebees can sense the electric field around a flower. Bees pick up a small electric charge as they fly, and flowers carry charge too. In the experiments, bees learned to tell charged artificial flowers from uncharged ones. They could even tell apart flowers whose electric fields had different shapes (Clarke, Whitney, Sutton and Robert, Science, 2013).

Think about what that means. The bee and the flower are not touching. There is only air between them. Yet each one pushes or pulls on the other.

You have met this puzzle before. The Earth pulls on the Moon across empty space. The gravitation lectures answered it with a field. This lecture builds the same idea for electric charge. Then it shows how to draw that field, because the picture will carry you through the whole topic.

A field is a property of space

Every charged object changes the space around it. Put another charge anywhere in that space, and it feels a force. We say that the first charge sets up an electric field.

The key point is this. The field is there whether or not a second charge is present to feel it. The source charge does not wait for a visitor. It has already changed every point around it. When a second charge arrives, it simply finds a force waiting at that spot.

So an electric field is a region where a charge feels a force. Notice the words "a charge". An uncharged object, such as a neutral ball of plastic far from anything, feels no electric force at all. The field acts only on charges.

Measuring a field: force per unit charge

How strong is the field at one point? The gravitation lectures measured the gravitational field with a small test mass. The field strength was the force on each kilogram: $g = F / m$ , in N/kg.

We do exactly the same here, with a small test charge instead of a test mass. Place a tiny charge q at the point. Measure the force F on it. The field strength is the force on each coulomb of charge.

Electric field strength

The electric field strength E at a point is the force per unit positive charge on a small test charge placed there.

$E = \frac{F}{q}$ , measured in newtons per coulomb, N/C.

The direction of the field is the direction of the force on a positive test charge.

Two words in that box need their reasons.

Why positive? A positive charge and a negative charge in the same place feel forces in opposite directions. If we did not choose one, the field would have no single direction. So everyone agrees to use a positive test charge. A negative charge then feels a force opposite to the field.

Why small? A test charge also has its own field. A large test charge would push the source charges around and change the very field we are trying to measure. So the test charge must be small enough that it disturbs nothing. A charge of a few microcoulombs or less is typical. A whole coulomb is enormous by comparison.

Once E is known, you can turn the definition around. The force on any charge q at that point is $F = qE$ . The next two lectures put numbers on E: Coulomb's law for a single charge, and a simple formula for the field between plates.

Worked Example 1 — Field strength from a force

A small positive charge of $4.0 \times 1 0^{- 6}$ C feels a force of $2.0 \times 1 0^{- 2}$ N, pointing east. Find the electric field at that point.

Think first. This is the definition: force per coulomb. The direction comes free, because the test charge is positive.

E = \frac{F}{q} = \frac{2.0 \times 1 0 ^{- 2}}{4.0 \times 1 0 ^{- 6}} = 5.0 \times 1 0^{3} N/C

The field is $5.0 \times 1 0^{3}$ N/C, pointing east.

Check. Each coulomb would feel 5000 N. Our charge is 4 millionths of a coulomb, so it feels $5000 \times 4.0 \times 1 0^{- 6} = 0.020$ N. ✓

Drawing the field: field lines

A field has a size and a direction at every point in space. That is far too much to write down. So physicists draw it. The pictures are made of field lines, an idea that grew from Michael Faraday's "lines of force". At every point on a field line, the line points in the direction of the field there.

There are four rules. Do not learn them as a list. Each one comes straight from the definition.

1. Lines point away from positive charges and towards negative charges. A positive test charge is pushed away from a positive charge and pulled towards a negative one, and the field direction is the force on a positive test charge.
2. Lines start on positive charges and end on negative charges. If there is no negative charge nearby, the lines carry on outwards, a long way off.
3. Lines never cross. At a crossing, the field would point in two directions at once. A test charge there would be pushed two ways, which is impossible. The field at each point has one direction.
4. Where lines are close together, the field is strong. Where they spread apart, it is weak. The number of lines from a charge is drawn in proportion to its size, so a charge of 2 gets twice as many lines as a charge of 1.

Rule 4 is a rough guide in a flat drawing. A real field fills three dimensions, and a page shows only one slice of it. So compare spacing to say "stronger here, weaker there", but do not measure it with a ruler.

One charge on its own

Around a single charge, the lines point straight out (positive) or straight in (negative), equally in every direction. This is called a radial field. The lines spread apart as they go outwards, so the field gets weaker with distance. The next lecture finds exactly how much weaker.

Top: the field of a positive charge points straight out. Bottom: the field of a negative charge points straight in. In both, the lines spread apart with distance, so the field gets weaker.

Two charges together

With two charges, the field at each point is the combination of both fields. Near each charge, its own field is strongest, so the lines look radial there. Further away, the two fields compete. Use the explorer to see what happens.

Explore — the field of two charges

The left charge is always +1. Choose the right charge. Then move the probe, a small positive test charge, along the line through both charges and read the force on it. Find a place where the force is zero, or where it changes direction.

Right charge:

Probe position:

These two pictures show the field of two equal positive charges, and of a positive and a negative charge of the same size. N marks the neutral point, where the field is zero.

Two equal positive charges. No line joins them. Midway between them, at N, the two pushes cancel.

A positive and a negative charge of the same size. Lines leave the positive charge and end on the negative one. Midway, the two fields point the same way and add up.

Look at what you found. With a positive and a negative charge, every point between them has a field pointing from the positive charge to the negative one. Both charges agree there, so the two fields add. At the midpoint, the field is exactly 2.00 units, twice what either charge gives alone.

With two equal positive charges, the midpoint is different. The left charge pushes the test charge right. The right charge pushes it left, equally hard. The two forces cancel, so the field there is zero. A point where the field is zero is called a neutral point. Notice also that no line joins the two positive charges. Lines only end on negative charges, so near the middle they turn away from each other.

With +1 and +2, the neutral point moves. It is no longer at the middle. It sits closer to the smaller charge. This is because the smaller charge's field is weaker, so the test charge must be nearer to it for the two pushes to balance.

Stop and think

A student draws the field of a positive and a negative charge, and adds one more line that goes straight from the negative charge to the positive one. What is wrong with that line?

Show the answer

It points the wrong way. Field lines show the force on a positive test charge, and that force points away from the positive charge and towards the negative one. So every line runs from + to −. A line from − to + would claim that a positive charge between them is pushed towards the positive charge, which never happens.

Two parallel plates: a uniform field

Now take two flat metal plates facing each other. Charge one plate positive and the other negative. Between them, the field is very simple.

Two parallel plates. Between them the lines are straight, parallel and evenly spaced: a uniform field. Near the ends of the plates the lines bulge outwards (the edge effect).

In the middle region, the lines go straight across, from the positive plate to the negative plate. They are parallel and evenly spaced. Equal spacing means equal strength. So the field has the same size and the same direction at every point. This is called a uniform field.

Look at the ends of the plates in the figure. There, the lines bulge outwards and spread apart, so the field is weaker and changes direction. This is called the edge effect. When a question says "the field between the plates is uniform", it means the region well away from the edges. Plates that are wide compared with the gap between them have a large uniform region.

Uniform fields are the workhorse of the topic. In a uniform field, a charge feels the same force everywhere. That makes uniform fields useful for steering beams of electrons and for holding tiny charged drops still, as the examples below show.

A field line is not a path

Here is a mistake that even good students make. Test yourself first.

Predict — does a charge follow a curved field line?

A small positive charge is held at rest on a curved field line between a positive and a negative charge. Then it is let go. What does it do? Choose one answer.

This figure shows a positive charge released from rest on a curved field line (solid). Its path (dashed) starts along the line, then swings to the outside of the curve.

A field line shows the direction of the force, and the force changes the velocity, not the position directly. A charge that starts at rest begins to move along the line, because that is the direction of the force. But as soon as it is moving, it carries that motion with it. On a curved line, it cannot turn as sharply as the line does, so it drifts to the outside of the curve. The two only agree when the lines are straight, as in a uniform field or a radial field.

This is the same idea as a car going round a bend. The steering (the force) points one way, but the car is already moving, so it needs time to turn.

More worked examples

Worked Example 2 — The force on an electron

An electron is in a uniform field of $5.0 \times 1 0^{3}$ N/C pointing east. The charge on an electron is $- 1.60 \times 1 0^{- 19}$ C. Find the force on it, and compare its acceleration with g. The mass of an electron is $9.11 \times 1 0^{- 31}$ kg.

Think first. Use F = qE for the size. Then remember that a negative charge is pushed against the field.

F = qE = 1.60 \times 1 0^{- 19} \times 5.0 \times 1 0^{3} = 8.0 \times 1 0^{- 16} N

The electron is negative, so the force points west, opposite to the field.

Its acceleration is $a = F / m = 8.0 \times 1 0^{- 16} \div 9.11 \times 1 0^{- 31} \approx 8.8 \times 1 0^{14}$ m/s². That is about $9 \times 1 0^{13}$ times g. For an electron, gravity is far too small to matter here.

Check. The force is tiny in newtons, but the mass is tinier still, so a huge acceleration makes sense. This is why electric fields can steer electron beams so easily.

Worked Example 3 — Holding a drop still

A tiny oil drop of mass $1.0 \times 1 0^{- 15}$ kg carries a negative charge of $3.2 \times 1 0^{- 19}$ C (two electrons' worth). It floats at rest between two horizontal plates. Find the size and direction of the field. Take g = 9.8 N/kg.

Think first. At rest means balanced forces. The electric force must point up, to balance the weight.

Weight: $m g = 1.0 \times 1 0^{- 15} \times 9.8 = 9.8 \times 1 0^{- 15}$ N, downwards.

Electric force needed: $9.8 \times 1 0^{- 15}$ N, upwards. So

E = \frac{F}{q} = \frac{9.8 \times 1 0 ^{- 15}}{3.2 \times 1 0 ^{- 19}} \approx 3.1 \times 1 0^{4} N/C

The drop is negative, so it is pushed against the field. The force is up, so the field points down. That means the top plate is positive and the bottom plate is negative.

Check. $qE = 3.2 \times 1 0^{- 19} \times 3.06 \times 1 0^{4} \approx 9.8 \times 1 0^{- 15}$ N, equal to the weight. ✓ Robert Millikan used drops held still in this way, in experiments published from 1909 to 1913, to measure the charge on an electron.

Worked Example 4 — Reasoning about a neutral point

Two positive charges, +1 and +4, are fixed a short distance apart. Where could the neutral point be: between them, or outside them? And is it nearer the +1 or the +4?

Think first. A neutral point needs two pushes that are equal in size and opposite in direction.

Outside the pair, on either side, both charges push the test charge the same way: away from both. Two pushes in the same direction cannot cancel. So there is no neutral point outside them.

Between them, the +1 pushes the test charge towards the +4, and the +4 pushes it back towards the +1. These are opposite, so they can cancel. The neutral point is between them.

The +4 has the stronger field at any given distance. To balance it, the test charge must be closer to the weaker +1. So the neutral point is nearer the +1.

Check. This matches the explorer with +1 and +2: the neutral point moved towards the smaller charge.

Three traps

Trap 1 — Lines joining two like charges

Field lines run from positive charges to negative charges. Between two positive charges, no line joins them: the lines bend away from each other, leaving a neutral point. Draw them meeting, and you are claiming that a field line runs from + to +.

Trap 2 — The force on a negative charge

The field direction is the force on a positive charge. An electron in the same field is pushed the opposite way. Always ask "is this charge positive or negative?" before you give a direction.

Trap 3 — Field lines as paths

A field line shows the direction of the force at each point, not the route a charge travels. A charge follows a line only when the line is straight. On a curved line, a moving charge swings to the outside of the curve.

Try it yourself

Work each one on paper first. Draw a sketch for every question and mark the direction of the field. Open a hint only when you are stuck.

A test charge of $+ 2.0 \times 1 0^{- 9}$ C feels a force of $6.0 \times 1 0^{- 5}$ N, pointing north. Find the electric field strength and its direction.

Hint 1
Field strength is force per coulomb.

Hint 2
Divide $6.0 \times 1 0^{- 5}$ by $2.0 \times 1 0^{- 9}$ . The charge is positive, so the field points the same way as the force.

Full answer
$E = F / q = 6.0 \times 1 0^{- 5} \div 2.0 \times 1 0^{- 9} = 3.0 \times 1 0^{4}$ N/C, pointing north (the test charge is positive). Check: $3.0 \times 1 0^{4} \times 2.0 \times 1 0^{- 9} = 6.0 \times 1 0^{- 5}$ N. ✓
An electron (charge $- 1.60 \times 1 0^{- 19}$ C) is in a field of $2.0 \times 1 0^{4}$ N/C pointing east. Find the size and direction of the force on it. What would change for a proton (charge $+ 1.60 \times 1 0^{- 19}$ C) at the same point?

Hint 1
Use F = qE for the size, using the size of the charge.

Hint 2
A negative charge is pushed against the field. A proton has the same size of charge but the opposite sign.

Full answer
$F = 1.60 \times 1 0^{- 19} \times 2.0 \times 1 0^{4} = 3.2 \times 1 0^{- 15}$ N. The electron is negative, so the force points west. A proton feels a force of the same size, $3.2 \times 1 0^{- 15}$ N, but pointing east, with the field. Check: same size of charge, same field, so same size of force; only the sign of the charge flips the direction. ✓
A dust grain of mass $4.0 \times 1 0^{- 14}$ kg carries a charge of $+ 8.0 \times 1 0^{- 19}$ C. It is held at rest by a vertical electric field. Find the size and direction of the field. Take g = 9.8 N/kg.

Hint 1
At rest means the electric force balances the weight. Which way must the electric force point?

Hint 2
Work out the weight mg, then divide by the charge. The grain is positive, so the field points the same way as the force on it.

Full answer
Weight $= 4.0 \times 1 0^{- 14} \times 9.8 = 3.92 \times 1 0^{- 13}$ N, down. The electric force must be $3.92 \times 1 0^{- 13}$ N, up. $E = 3.92 \times 1 0^{- 13} \div 8.0 \times 1 0^{- 19} = 4.9 \times 1 0^{5}$ N/C. The grain is positive, so the field points up, the same way as the force. Check: $8.0 \times 1 0^{- 19} \times 4.9 \times 1 0^{5} = 3.92 \times 1 0^{- 13}$ N, equal to the weight. ✓
A positive and a negative charge of the same size are fixed a short distance apart. Is there a neutral point anywhere on the line through them (other than very far away)? Explain using the directions of the two pushes.

Hint 1
Look at three regions: between the charges, beyond the positive one, and beyond the negative one. In each, which way does each charge push a positive test charge?

Hint 2
Outside the pair, the two pushes are opposite. But which charge is nearer, and which field is stronger there?

Full answer
No. Between the charges, the positive charge pushes the test charge towards the negative one, and the negative one pulls it the same way. The pushes add, so they cannot cancel. Beyond either charge, the two pushes are opposite. But the test charge is always nearer to one of them, and since the charges are the same size, the nearer one has the stronger field. So the pushes are never equal, and there is no neutral point. Check with the explorer: choose −1 and move the probe everywhere. The force is never zero.

The bigger idea

Faraday began drawing "lines of force" in the 1830s, from the patterns that iron filings make around magnets. He used the word "field" in 1845, and set out his idea of lines of force fully in papers of 1851 and 1852. Many physicists of his time saw the lines as a drawing aid, nothing more. Faraday came to believe they described something real in the space itself.

James Clerk Maxwell turned Faraday's pictures into mathematics, starting with a paper in 1855–56. That work led him to light as a wave of electric and magnetic fields, the idea at the heart of the electromagnetic spectrum lectures. So when you draw a field line, you are using the same picture that led to the discovery of what light is. The field is not just a way of calculating forces. It is something real, spread through space, that can even travel on its own.

Key concepts in this lesson

An electric field is a region where a charge feels a force, and it exists around a charged object whether or not another charge is there to feel it.
Electric field strength is the force per unit positive charge on a small test charge, $E = F / q$ , in N/C, and its direction is the force on a positive charge.
A negative charge feels a force opposite to the field, and the force on any charge is $F = qE$ .
Field lines point from positive to negative, never cross, and are closest together where the field is strongest.
Between parallel plates the field is uniform away from the edges, a neutral point appears where two pushes cancel, and a field line is not the path a charge follows.