The Electric Field and Its Lines

Force per unit charge

Some fixed charges sit somewhere nearby. Put a small charge \(q_0\) at a point P and measure the force on it, say \(0.01\,\text{N}\) for \(q_0 = 2\,\mu\text{C}\). Double the charge at P and the force doubles; halve it and the force halves. The ratio \(F/q_0\) stays the same. That ratio belongs to the point P and to the charges around it, not to the charge we placed there.

That ratio is the electric field \(\vec E\) at P. The small charge used to find it is a test charge. Here \(E\) \(= 0.01/2 \times 10^{-6}\) \(= 5000\,\text{N/C}\). The field is a vector. By convention it points along the force on a positive test charge.

Once the field at P is known, the force on any charge there follows at once: \(\vec F = q\vec E\). A \(-4\,\mu\text{C}\) charge at our point P feels \(4 \times 10^{-6} \times 5000\) \(= 0.02\,\text{N}\). Its charge is negative, so the force points opposite to \(\vec E\). This is the use of the field. Work out the field of the sources once, then read off the force on any charge you place.

The field is the same idea as the gravitational field \(\vec g = \vec F/m\) of the chapter on Gravitational Force and Field. There the test body was a mass; here it is a charge. One difference follows from the two signs of charge. A gravitational field pulls every mass the same way, but an electric field pushes positive and negative charges in opposite directions.

The field of a point charge

Put a test charge \(q_0\) a distance \(r\) from a point charge \(q\). Coulomb's law gives a force of size \(kqq_0/r^2\). Divide by \(q_0\), and the test charge drops out. What remains is the field of the point charge. It points straight away from a positive charge and straight towards a negative one.

Fields add as vectors

Forces obey superposition, and the field is force per unit charge, so fields obey it too. At any point, the field of many charges is the vector sum of the fields of each: \(\vec E\) \(= \vec E_1 + \vec E_2 + \cdots\). Each part points away from its own charge if that charge is positive, towards it if negative.

Set the test charge at the midpoint, \(x = 0\) and \(y = 0\), with unlike charges. Each charge is \(10\,\text{cm}\) away and makes \(9 \times 10^9 \times 2 \times 10^{-6}/0.01\) \(= 1.8 \times 10^6\,\text{N/C}\). The field of \(+2\,\mu\text{C}\) points away from it, towards the negative charge. The field of \(-2\,\mu\text{C}\) points towards itself, the same way. So they add to \(3.6 \times 10^6\,\text{N/C}\), from the positive charge to the negative one. Switch to like charges and the two fields point opposite ways and cancel: the field at the midpoint is zero.

Move the test charge around and watch the triangle of arrows. Close to one charge its own field swamps the other. Off the line joining the charges, the two fields point at an angle. Only their vector sum tells where a positive charge would be pushed.

Drawing the field with lines

A field is an arrow at every point, far too many to draw. Faraday drew curves instead. Start at a positive charge and move always in the direction of the field there. The curve you trace is a field line. Its tangent at each point shows the direction of \(\vec E\). Draw many such lines and the whole field can be seen at a glance.

Look through the chips of Fig 4.1. Around one positive charge the lines run straight out, like spokes. Near the charge they are crowded; far away they spread. The field there is strong near the charge and weak far away. This is no accident. The number of lines is fixed, while the sphere they cross has area \(4\pi r^2\). So the lines per square metre fall as \(1/r^2\), exactly like \(kq/r^2\). The crowding of the lines shows the size of the field.

The patterns follow a few rules.

• A line starts on a positive charge and ends on a negative charge, or runs off to a great distance.
• The number of lines leaving or ending on a charge is proportional to the charge.
• The tangent to a line gives the direction of \(\vec E\), and the crowding of lines gives its size.
• Two lines never cross, because the field has only one direction at each point.
• Electrostatic field lines never form closed loops.
• In a uniform field the lines are straight, parallel and evenly spaced.

For two like charges the lines push away from the space between them, and no line crosses the midpoint. For a positive and an equal negative charge, every line from one ends on the other. For \(+2q\) and \(-q\), half the lines from \(+2q\) end on \(-q\) and half escape.

A closed loop is impossible for a reason of energy. Carry a positive charge once round such a loop, always along the field. The field would do positive work on every step and return the charge to its start with energy gained from nothing. The next chapter, Electric Potential, Energy and the Dipole, shows why the electrostatic field never allows that.

A charge in a uniform field

Between two large parallel plates with opposite charges, the field is the same at every point. Such a field is a uniform field. A charge \(q\) of mass \(m\) there feels the same force \(qE\) wherever it goes. A constant force gives a constant acceleration, \(a = qE/m\). So the equations of motion from the chapter on Motion in One Dimension apply directly.

In Fig 4.2 the force on the particle is \(qE\) \(= 2 \times 10^{-6} \times 5000\) \(= 0.01\,\text{N}\), so \(a = 0.01/0.001 = 10\,\text{m/s}^2\). From rest, \(s = \tfrac12 at^2\) gives \(20 = 5t^2\), so the particle covers \(20\,\text{m}\) in \(2\,\text{s}\). By then its speed is \(at = 20\,\text{m/s}\). A negative particle of the same mass and charge size does exactly the same, but against the field.

An electron shows how light a charge can be. In the same \(5000\,\text{N/C}\), its acceleration is \(eE/m_e\) \(= 1.6 \times 10^{-19} \times 5000/9.1 \times 10^{-31}\) \(= 8.8 \times 10^{14}\,\text{m/s}^2\). From rest it crosses \(1\,\text{cm}\) in about \(5\,\text{ns}\) and reaches \(4.2 \times 10^6\,\text{m/s}\). The electric force on it is about \(10^{14}\) times its weight. So gravity can be ignored for electrons and ions in any laboratory field.

The spark from a sweater

Now the crackle of the winter sweater from lesson 1 can be explained. Rubbing leaves large charges on the sweater and on your shirt, close together. Near sharp folds and fibres the field grows very strong. Dry air is an insulator, but only up to about \(3 \times 10^6\,\text{N/C}\). Beyond that, the field tears electrons from air molecules, and the air conducts in a sudden spark. This is dielectric breakdown of the air, and each tiny spark crackles and glows blue.