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“Electric Charges and Fields”

 

 

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Class XII · Chapter 01

Electric Charges and Fields

The foundation of electrostatics — how charge behaves, why Coulomb’s Law works the way it does, and how the electric field lets us describe force without contact.

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⏱ 22 min read
🎯 34 practice questions
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1Electric Charge

Electric charge is a basic property of matter, alongside mass, that determines how a particle interacts electrically and magnetically with everything around it. Rub a glass rod with silk, or a plastic comb through dry hair, and you can pick up small bits of paper — that’s charge in action, first noticed in ancient Greece with amber (“elektron”), which is where the word “electricity” comes from.

Benjamin Franklin named the two kinds of charge positive and negative. Charge on a glass rod rubbed with silk is called positive; charge on a plastic rod rubbed with fur is called negative. The rule that follows from experiment:

  • Like charges repel each other.
  • Unlike charges attract each other.

Conductors and Insulators

Materials differ in how easily charge moves through them. In conductors (metals, the human body, earth), electrons move relatively freely, so charge given at one point spreads over the surface almost instantly. In insulators (glass, rubber, most plastics), charge stays localized where it’s placed, since electrons are tightly bound to atoms.

Charging by Induction

You can charge a conductor without touching it: bring a charged rod near an uncharged conductor, and charges within the conductor redistribute — opposite charge accumulates on the near side, like charge on the far side. Ground the far side momentarily and disconnect, and the conductor is left with a net charge opposite to the rod’s — without ever transferring charge directly.

2Basic Properties of Electric Charge

Additivity of Charge

Charges add up like real numbers — if a system has charges q₁, q₂, q₃, … at different points, the total charge is simply their algebraic sum, taking sign into account.

Conservation of Charge

The total charge of an isolated system never changes. Charge isn’t created or destroyed — it’s only transferred from one body to another. When you rub glass with silk, the glass doesn’t gain charge from nowhere; electrons move from glass to silk (or vice versa), so the combined charge of both stays exactly zero, same as before.

Quantization of Charge

Charge doesn’t come in arbitrary amounts — it always exists as an integer multiple of a smallest unit, the charge of an electron.

Quantization

q = n e
where n is an integer (positive or negative) and e = 1.6 × 10⁻¹⁹ C is the elementary charge. At the macroscopic scale — where n runs into billions — charge appears continuous, the same way a beach looks like continuous sand rather than individual grains.

3Coulomb’s Law

Coulomb’s Law gives the force between two point charges at rest, separated by some distance. It’s the electrostatic equivalent of Newton’s law of gravitation, and it’s the starting point for almost everything else in this chapter.

Coulomb’s Law (magnitude)

F = k · q₁q₂ / r²
k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²/C², where ε₀ (permittivity of free space) ≈ 8.854 × 10⁻¹² C²/N·m². The force acts along the line joining the two charges — attractive if the charges are unlike, repulsive if they’re alike.

In vector form, the force on charge q₁ due to q₂, at position vectors r₁ and r₂, is written using the unit vector r̂₂₁ pointing from q₂ to q₁:

Vector Form

F₂₁ = k · q₁q₂ / r² · r̂₂₁
This form automatically handles direction — plug in the signs of q₁ and q₂ and the vector works out attractive or repulsive on its own.
Common Slip
Forgetting that Coulomb’s Law only applies exactly to point charges (or spherically symmetric charge distributions, treated as concentrated at the centre). For irregular or extended shapes, you need to integrate over the distribution instead.

4Forces Between Multiple Charges: Superposition

When more than two charges are present, the net force on any one charge is simply the vector sum of the forces due to each of the other charges individually, as if the others weren’t there. This is the superposition principle, and it’s what makes electrostatics tractable — you can always break a complicated arrangement down into pairwise Coulomb forces and add them as vectors.

Superposition

F₁ = F₁₂ + F₁₃ + F₁₄ + …
Each term is computed independently via Coulomb’s Law, then added head-to-tail as vectors — not as scalars.

5Electric Field

Rather than tracking forces between every possible pair of charges, it’s more useful to ask: what does a charge do to the space around it? The answer is the electric field — a vector quantity defined at every point in space, describing the force a small positive “test charge” would feel if placed there.

Definition

E = F / q₀
where q₀ is a vanishingly small positive test charge, so its own field doesn’t disturb the source charges being measured. Units: N/C, equivalently V/m.

The field due to a single point charge Q at distance r follows directly from Coulomb’s Law:

Field of a Point Charge

E = k Q / r²
Pointing radially outward if Q is positive, radially inward if Q is negative.

6Electric Field Lines

Field lines are a way to visualize E — imaginary curves drawn so the tangent at any point gives the field’s direction there, and the density of lines indicates field strength.

  • Field lines start on positive charges and end on negative charges (or go off to infinity).
  • Two field lines never cross — if they did, the field would have two directions at that point, which is meaningless.
  • Lines are denser where the field is stronger, and spread out where it’s weaker.
  • In a uniform field, field lines are straight, parallel, and equally spaced.

7Electric Dipole

An electric dipole is a pair of equal and opposite charges (+q and −q) separated by a small distance 2a. Dipoles matter well beyond textbook problems — polar molecules like water behave as tiny electric dipoles, which is central to a lot of chemistry and biology.

Dipole Moment

p = q · (2a)
A vector pointing from the negative charge to the positive charge. Unit: C·m.

Field on the Axial Line

Along the line through both charges, extended outward, for a point far from the dipole (r ≫ a):

Axial Field

Eaxial = 2kp / r³
Direction: same as p (along the dipole axis, pointing from −q side to +q side).

Field on the Equatorial Line

Along the perpendicular bisector of the dipole, for r ≫ a:

Equatorial Field

Eequatorial = kp / r³
Direction: opposite to p, and exactly half the magnitude of the axial field at the same distance — a frequently tested comparison.

Dipole in a Uniform External Field

A uniform field exerts no net force on a dipole (the forces on +q and −q are equal and opposite), but it does exert a torque that tries to align the dipole with the field:

Torque

τ = p E sin θ
where θ is the angle between p and E. Torque is zero when the dipole is aligned with (θ=0°) or against (θ=180°) the field, and maximum when perpendicular (θ=90°).

8Electric Flux

Electric flux is a measure of how much field “passes through” a given surface — think of it loosely as counting field lines crossing an area.

Flux Through a Flat Surface

Φ = E · A = EA cos θ
where θ is the angle between the field E and the area vector A (normal to the surface). Unit: N·m²/C. For a curved or non-uniform case, flux is found by integrating E·dA over the whole surface.

9Gauss’s Law

Gauss’s Law connects the electric flux through any closed surface to the total charge enclosed by it — and it’s exact, not an approximation, following directly from Coulomb’s Law.

Gauss’s Law

Φ = qenclosed / ε₀
The total flux through a closed surface depends only on the net charge inside — the shape of the surface and the position of the charge within it don’t matter, and charges outside the surface contribute nothing to the net flux.
Why It’s Powerful
For any charge distribution with enough symmetry (spherical, cylindrical, or planar), Gauss’s Law lets you find E algebraically, by choosing a “Gaussian surface” that matches the symmetry — sidestepping messy integration entirely.

10Applications of Gauss’s Law

Infinite Line Charge

Field at distance r from the line

E = λ / (2πε₀r)
λ is linear charge density (C/m). Use a cylindrical Gaussian surface coaxial with the line.

Infinite Plane Sheet of Charge

Field near an infinite sheet

E = σ / (2ε₀)
σ is surface charge density (C/m²). Notably, this field is uniform — it doesn’t fall off with distance from the sheet.

Uniformly Charged Thin Spherical Shell

Outside the shell (r ≥ R)

E = kQ / r²
Identical to the field of a point charge Q at the centre.
Inside the shell (r < R)

E = 0
A classic and often-tested result: the field is exactly zero everywhere inside a uniformly charged shell.

Formula Summary

Quantization
q = ne
Coulomb’s Law
F = kq₁q₂/r²
Electric Field
E = F/q₀ = kQ/r²
Dipole Moment
p = q(2a)
Axial Field
E = 2kp/r³
Equatorial Field
E = kp/r³
Torque on Dipole
τ = pE sinθ
Electric Flux
Φ = EA cosθ
Gauss’s Law
Φ = q/ε₀
Infinite Sheet
E = σ/2ε₀

Solved Examples

Example 1 · Coulomb’s LawTwo point charges, +3 μC and −3 μC, are placed 30 cm apart in air. Find the force between them.

Solution: F = kq₁q₂/r² = (9×10⁹ × 3×10⁻⁶ × 3×10⁻⁶) / (0.3)² = (8.1×10⁻²) / (0.09) = 0.9 N.

Since the charges are unlike, the force is attractive, directed along the line joining them.

Example 2 · Electric DipoleA dipole with moment p = 4×10⁻⁹ C·m is placed at a point 0.1 m from its centre on the equatorial line. Find the electric field there.

Solution: E = kp/r³ = (9×10⁹ × 4×10⁻⁹) / (0.1)³ = 36 / 0.001 = 3.6×10⁴ N/C, directed opposite to p.

Quick Check

1. If the distance between two point charges is doubled, the Coulomb force between them becomes:
Twice
Half
One-fourth
✓ One-fourth (correct)
2. The electric field inside a uniformly charged spherical shell is:
✓ Zero (correct)
Maximum
Equal to the field on the surface
Depends on r
Up Next · Chapter 02

Electrostatic Potential and Capacitance

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