Electric Charges and Fields: From Coulomb to Gauss
A from-scratch guide to charge, Coulomb's law, the electric field and its lines, the dipole, and Gauss's law -- with a worked numerical, a full derivation, and a quick-reference table for CBSE Class 12 (and your JEE/NEET/CUET foundation).
By the PadhoDost Team · 📖 8 min read · Updated 4 August 2026
Part of Class 12 (CBSE) prep🧠 The winter sweater crackle
On a dry winter morning, pull a woollen sweater over your head and you hear a tiny crackle -- sometimes even a faint spark in the dark. Rub a plastic comb on your hair and it lifts little bits of paper. No magic is involved: you are simply rubbing tiny electrons from one surface to another, leaving things electrically charged. This whole chapter is about turning that everyday crackle into precise, powerful laws. Padho, dost -- let's begin!
Electric Charge: the basics
Charge is a basic property of matter, just like mass. There are two kinds -- positive and negative -- and the rule is simple: like charges repel, unlike charges attract. A body becomes charged only when electrons are transferred: lose electrons and it turns positive, gain electrons and it turns negative. Charge is measured in coulombs (C), the SI unit. One coulomb is actually a huge amount -- the charge you feel from a rubbed comb is only a few billionths of a coulomb.
Three properties you must remember
- ✓Quantisation: charge comes in whole packets. q = ne, where e = 1.6 x 10^-19 C and n is an integer (0, +-1, +-2, ...). You can never have half an electron's worth of charge.
- ✓Conservation: the total charge of an isolated system never changes. Charge is only transferred, never created or destroyed.
- ✓Additivity: the net charge of a body is the ordinary algebraic sum of all the charges on it (a +3 C and a -1 C together give a net +2 C).
- ✓SI unit is the coulomb (C); like charges repel, unlike charges attract.
Coulomb's Law -- how strong is the pull or push?
Coulomb's law says the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. It is a vector, directed along the line joining the charges. Notice how similar it looks to Newton's law of gravitation, but with one big difference: gravity only pulls, while the electric force can pull (unlike charges) or push (like charges).
📝 Worked example: force between two charges
Two point charges q1 = +2 uC and q2 = +3 uC are 30 cm apart in vacuum. Find the force between them.
Convert to SI units: q1 = 2 x 10^-6 C, q2 = 3 x 10^-6 C, r = 0.30 m.
Write Coulomb's law: F = (1/4 pi e0) . q1 q2 / r^2 = 9 x 10^9 x (q1 q2) / r^2.
Numerator: q1 q2 = (2 x 10^-6)(3 x 10^-6) = 6 x 10^-12 C^2.
Denominator: r^2 = (0.30)^2 = 0.09 m^2.
Substitute: F = 9 x 10^9 x (6 x 10^-12 / 0.09) = 9 x 10^9 x 6.67 x 10^-11 = 0.6 N.
Both charges are positive, so the force is repulsive: 0.6 N on each charge, pushing them apart along the line joining them.
Electric Field and Field Lines
Instead of talking about force every time, physicists say a charge fills the space around it with an electric field. The electric field E at a point is the force felt by a tiny, positive test charge q0 placed there, per unit charge: E = F/q0. It is a vector with SI unit N/C (also written V/m). For a single point charge q, the field at distance r is E = (1/4 pi e0) . q/r^2, pointing away from a positive charge and towards a negative one. When many charges are present, just add their fields as vectors -- this is the superposition principle.
How to read electric field lines
- ✓They start on positive charges and end on negative charges (or run off to infinity).
- ✓The tangent to a line at any point gives the direction of E there.
- ✓Where lines are crowded, the field is strong; where they spread out, it is weak.
- ✓Two field lines can never cross -- the field has only one direction at each point.
- ✓In electrostatics they never form closed loops.
The Electric Dipole
Gauss's Law -- a shortcut for symmetric charges
Deriving the field of an infinite charged sheet
- 1Take a large flat sheet with uniform surface charge density sigma (charge per unit area). By symmetry the field E must point straight out, perpendicular to the sheet, with the same strength on both sides.
- 2Choose a Gaussian surface: a small cylinder (pillbox) of cross-section area A that pierces the sheet, with its two flat faces parallel to the sheet, one on each side.
- 3Flux through the curved side = 0, because E is parallel to that surface. Flux through each flat face = E x A, so the total flux = 2EA.
- 4Charge enclosed by the pillbox = sigma x A (only the patch of sheet inside it).
- 5Apply Gauss's law: 2EA = sigma A / e0. The area A cancels, giving E = sigma / (2 e0).
- 6Result: the field of an infinite sheet is uniform, E = sigma/(2 e0), and does not depend on the distance from the sheet.
| Charge distribution | Field magnitude E | Direction |
|---|---|---|
| Point charge q | q / (4 pi e0 r^2) | Radial, away from +q |
| Infinite line charge, lambda | lambda / (2 pi e0 r) | Radial, perpendicular to the line |
| Infinite plane sheet, sigma | sigma / (2 e0) | Perpendicular to sheet, uniform |
| Spherical shell (outside), Q | Q / (4 pi e0 r^2) | Radial, acts like a point charge at centre |
| Spherical shell (inside) | 0 | No field inside |
⚡ Quick check
A single point charge +q is placed exactly at the centre of a cube. What is the electric flux through just one face of the cube?
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