Electric Charges and Fields

Written and reviewed by Akanksha Tiwari, Physics & Chemistry faculty at UniDrill (M.Sc. Physics, Banaras Hindu University · 8 years teaching Class XI–XII science). Based on NCERT Physics Class XII, Chapter 1.

Why does a charged comb pick up paper?

Run a plastic comb through dry hair and hold it over tiny bits of paper. They jump up and stick to it. But the paper is neutral, so why should a charged comb attract it? The answer is the whole chapter in miniature. The comb creates an electric field around itself. That field nudges the charges inside each paper molecule, pulling the nearer side of the paper slightly opposite in sign to the comb and pushing the far side slightly the same sign. The near charge feels a stronger pull because the field is stronger closer to the comb, so the paper is attracted. By the end of this chapter you will be able to explain each step with a formula.

Charge: what it is and what it can do

There are two kinds of electric charge, called positive and negative. Like charges repel and unlike charges attract. A glass rod rubbed with silk becomes positive, and a plastic rod rubbed with fur becomes negative. In every ordinary case of charging, only electrons move from one body to the other. The body that loses electrons becomes positive, and the one that gains them becomes negative.

Materials differ in how freely they let charge move.

Charge has three basic properties.

Millikan's oil drop experiment (1909–1912) showed that the charge on tiny oil droplets always came in small whole number multiples of e. At everyday scales the unit is so small that charge seems to flow smoothly. A charge of 1 µC contains more than $10^{12}$ electrons.

Beyond the textbook. For the exam: e is the smallest free charge, so charges are integral multiples of e. In reality: quarks, the particles inside protons and neutrons, carry charges of $\pm e/3$ and $\pm 2e/3$. They are never found free, because the strong force always keeps them bound in groups whose total charge is a whole number of e. Why it matters: quantisation is true of everything we can isolate and measure, which is why the exam rule holds.

Worked example: counting electrons. A plastic rod gains a net charge of −4.8 µC by rubbing. How many electrons were transferred to it, and what is their total mass? (e = $1.6 \times 10^{-19}$ C, mass of an electron $9.1 \times 10^{-31}$ kg.)

The common wrong step: forgetting to convert µC to C and getting $3 \times 10^{19}$. The total mass of about $3 \times 10^{-17}$ kg shows how tiny a transfer produces a noticeable charge.

Coulomb's law: the force between two point charges

For two point charges $q_{1}$ and $q_{2}$ at distance r in vacuum, the force acts along the line joining them. It is repulsive if the charges have the same sign and attractive if they have opposite signs. Its magnitude is

$$F = \dfrac{k\,q_{1}q_{2}}{r^{2}} = \dfrac{1}{4\pi\varepsilon_{0}}\,\dfrac{q_{1}q_{2}}{r^{2}}$$

with $k = 9 \times 10^{9}$ N m² C⁻² and $\varepsilon_{0} = 8.854 \times 10^{-12}$ C² N⁻¹ m⁻². Never confuse the two: k is huge and $\varepsilon_{0}$ is tiny, and one is the reciprocal of $4\pi$ times the other. In vector form, the force on charge 2 due to charge 1 is $\vec{F}{21} = k\,q{1}q_{2}\,\hat{r}{21}/r^{2}$, where $\hat{r}{21}$ is the unit vector pointing from 1 to 2. Newton's third law holds, $F_{12} = -F_{21}$. Coulomb's law is valid from nuclear distances (about $10^{-15}$ m) up to very large distances.

Coulomb's law looks like Newton's law of gravitation, but there are important differences.

That last row explains why planets, though made of enormous numbers of charges, are held together by gravity: matter is electrically neutral overall, so the electric forces cancel and the weak gravity remains.

Superposition. The force on a charge due to several other charges is the vector sum of the forces due to each one separately. The presence of other charges does not alter the force between any pair. Coulomb's law plus superposition is enough to build the whole of electrostatics.

Beyond the textbook. For the exam: all forces in this chapter are in vacuum (or air), with the constant k = $9 \times 10^{9}$. In reality: in a medium of dielectric constant K the force between the same charges is K times smaller. Water has K of about 80, so the attraction between Na⁺ and Cl⁻ ions drops to about 1/80 of its value in air. Why it matters: that is why salt dissolves in water so easily, and you will meet K again in the next chapter.

Worked example: force on a charge in a triangle. Three equal charges of +2 µC sit at the corners of an equilateral triangle of side 0.3 m. Find the net force on one of them.

The common wrong step: adding the magnitudes, 0.4 + 0.4 = 0.8 N. Forces are vectors, and these two are 60° apart.

Worked example: where is the force zero? Charges +4 µC and +1 µC are 30 cm apart in air. Find the point on the line joining them where a third charge would feel no net force.

The common wrong step: choosing the midpoint. The null point is always closer to the smaller charge. Another useful result follows: if three free charges are all in equilibrium on a line, the middle one has the sign opposite to the two outer ones. They can never all have the same sign.

The electric field

Charge A pulling on charge B across empty space feels like action at a distance. The field idea removes the mystery. Each charge changes the space around it, and any other charge placed there responds to the field at its own location. The electric field at a point is the force per unit positive test charge, with the test charge small enough not to disturb the source:

$E = F/q$ (as $q \to 0$), unit N C⁻¹ (equal to V m⁻¹)

For a point charge Q, $E = kQ/r^{2}$, directed away from Q if it is positive and towards Q if negative. The force on any charge q in a field is $F = qE$, along E for a positive charge and opposite for a negative one. For several sources, the net field is the vector sum of the fields of each.

Worked example: field at the midpoint of a pair. Charges +$10^{-8}$ C and −$10^{-8}$ C are 0.1 m apart. Find the field at the midpoint and the force on an electron placed there.

The common wrong step: cancelling the two fields because the charges are equal and opposite. They cancel at the midpoint only for two like charges.

Field lines. Faraday's picture draws curves whose tangent gives the direction of E. The density of lines (lines per unit area at right angles) shows the strength. Field lines start on positive charges and end on negative charges (or go to infinity). They never cross, because E has one direction at each point. In electrostatics they never form closed loops. The lines of a point charge thin out as $1/r^{2}$, matching the inverse-square law.

The electric dipole

A dipole is a pair of equal and opposite charges +q and −q separated by a small distance 2a. Its dipole moment is

$p = q \times 2a$, directed from −q to +q, unit C m

At a distance r much greater than 2a, a dipole's field is much weaker than a single charge's, because the two opposite fields almost cancel. It falls as $1/r^{3}$ rather than $1/r^{2}$.

So at the same distance, the axial field is exactly twice the equatorial field.

In a uniform field. The two charges feel equal and opposite forces, so the net force is zero. But the forces act at different points, giving a torque

$\tau = p \times E$, magnitude $\tau = pE\sin\theta$

where $\theta$ is the angle between p and E. The torque turns the dipole towards alignment with the field. It is maximum at $\theta = 90^{\circ}$ and zero at $\theta = 0^{\circ}$ or $180^{\circ}$.

Pause and predict: a dipole is placed in a non-uniform field, stronger on its left. Is the net force zero?

Check your answer: no. The charge on the stronger side feels the larger force, so there is a net force on the dipole towards the stronger part of the field (for a dipole aligned with the field). This is exactly why a charged comb attracts a neutral paper bit: the paper's induced dipole is pulled towards where the comb's field is strongest.

Beyond the textbook. For the exam: a dipole is two point charges a short distance apart. In reality: many molecules are permanent dipoles. A water molecule has p of about $6 \times 10^{-30}$ C m, because its oxygen end is slightly negative and its hydrogen end slightly positive. Why it matters: the rotating torque on water molecules in an oscillating field is how a microwave oven heats food, and the large dipole moment is why water is such a good solvent.

Worked example: a dipole in a field. Two charges +5 µC and −5 µC are 2 mm apart. Find (a) its dipole moment, (b) the field it produces at 0.5 m on the axis and on the equatorial line, and (c) the torque when it is placed at 30° to a uniform field of $5 \times 10^{4}$ N C⁻¹.

The common wrong step: treating the $1/r^{2}$ point charge formula as the dipole field, or using $\sin\theta$ with $\theta$ measured from the perpendicular to the field instead of from the field itself.

Electric flux and Gauss's law

Flux measures how much field passes through a surface. For a small flat area $\Delta S$, taken as a vector pointing along its outward normal, the flux is $\Delta\Phi = E \cdot \Delta S = E\,\Delta S \cos\theta$. For a bigger surface add up the pieces: $\Phi = \int E \cdot dS$. Its unit is N m² C⁻¹. Flux is positive where the field leaves a closed surface, negative where it enters.

Gauss's law states that the net flux through any closed surface is the charge enclosed by it, divided by $\varepsilon_{0}$:

$\Phi = \oint E \cdot dS = q/\varepsilon_{0}$, where q is the net charge inside the surface

Three things to note. The flux depends only on the charge inside, not on its position or on the shape of the surface. Charges outside contribute nothing to the net flux, because their field lines enter the surface and leave it again. But the E in the integral is the field due to all charges, inside and outside. And the law is a consequence of the inverse-square form of Coulomb's law. The area of a sphere grows as $r^{2}$ and the field falls as $1/r^{2}$, so the flux through every enclosing sphere is the same. With any other power of r this would fail.

For a point charge q at the centre of a cube, the total flux is $q/\varepsilon_{0}$. By symmetry each of the six faces gets one sixth, $q/6\varepsilon_{0}$.

Pause and predict: a point charge q sits exactly at one corner of a cube. What is the flux through the entire cube?

Check your answer: $q/8\varepsilon_{0}$. Imagine eight identical cubes meeting at the charge. Together they surround it completely, so they share $q/\varepsilon_{0}$ equally, one eighth each.

Using symmetry. When the charge distribution is symmetrical, choose a Gaussian surface on which E is constant in magnitude and either perpendicular or parallel to the surface. Then the integral becomes simple.

Two parallel sheets with $+\sigma$ and $-\sigma$ give a uniform field $\sigma/\varepsilon_{0}$ in the region between them and zero outside, because the fields of the two sheets add between them and cancel outside. The field just outside a charged conductor is also $\sigma/\varepsilon_{0}$. The reason is that the conductor's own surface charge produces $\sigma/2\varepsilon_{0}$, and the rest of the charge adds another $\sigma/2\varepsilon_{0}$ outside the conductor while cancelling inside it.

Worked example: three standard fields. Find the field (a) 20 cm from a long straight wire with $\lambda = 4 \times 10^{-9}$ C m⁻¹, (b) near a large sheet with $\sigma = 2 \times 10^{-6}$ C m⁻², and (c) at 30 cm and at 5 cm from the centre of a thin shell of radius 10 cm carrying 5 µC.

The common wrong step: using $\sigma/\varepsilon_{0}$ for an isolated sheet, or assuming the field inside a charged shell is $kq/R^{2}$. The field is zero for r < R.

Exam traps

Formula sheet

60-second revision

Where this chapter connects

• Before this: Laws of Motion and Gravitation give you the habits of force vectors and inverse-square laws that Coulomb's law reuses.
• Next: Electrostatic Potential and Capacitance turns the field into the scalar potential and puts it to work in capacitors.
• Forward link: Current Electricity is what you get when charges move through a conductor, and Moving Charges and Magnetism shows what moving charges do to each other.