Class 12 · Chapter 1

Electric Charges & Fields

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Electric Charges & Fields Formula Sheet

8 formulas across 3 topics in Electric Charges & Fields.

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Updated 2026-07-05 · v1.0.0

Coulomb's Law & Electric Field

Coulomb's law is the electrostatic twin of gravitation — inverse square, but with two signs and vastly stronger. Superpose forces or fields as vectors, always.

F = (1/4πε₀) q₁q₂/r² = k q₁q₂/r²

Coulomb's law

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
k	Coulomb constant 1/4πε₀k = 9 × 10⁹ N·m²/C²	N·m²/C²	[ML³T⁻⁴A⁻²]
ε₀	permittivity of free space8.85 × 10⁻¹² C²/(N·m²)	C²/(N·m²)	[M⁻¹L⁻³T⁴A²]
q₁, q₂	the two point charges	C	[AT]
r	separation between charges	m	[L]

Valid when

Point charges (or spherical charges, centre-to-centre)
In a medium of dielectric constant K, divide by K

Common mistakes

Coulomb force obeys superposition — the force between two charges is unaffected by a third; add the separate forces as vectors

Read the concept →

E = F/q₀ = (1/4πε₀) q/r²

Electric field of a point charge

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
E	electric field strength	N/C (= V/m)	[MLT⁻³A⁻¹]
q₀	small positive test charge	C	[AT]
q	source charge	C	[AT]

Valid when

Field points AWAY from positive charge, TOWARD negative

Common mistakes

E is defined in the LIMIT of a vanishing test charge, so the test charge does not disturb the source

Read the concept →

Worth remembering

Field lines start on positive, end on negative charge; they never cross, and their density shows field strength
Electrostatic force is ~10³⁹ times stronger than gravity between two protons

Electric Dipole

A dipole is two equal and opposite charges a short distance apart. Its field falls as 1/r³ — faster than a point charge — and it feels torque in a uniform field.

p = q(2a), directed from −q to +q

Electric dipole moment

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
p	dipole moment	C·m	[ATL]
2a	separation between the charges	m	[L]

Valid when

Convention: points from negative to positive charge

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E(axial) = (1/4πε₀) 2p/r³; E(equatorial) = (1/4πε₀) p/r³

Dipole field — axial and equatorial

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Superposing the fields of +q and −q for r ≫ a

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
r	distance from the dipole centre	m	[L]
p	dipole moment	C·m	[ATL]

Valid when

Short dipole approximation r ≫ a
Axial field is TWICE the equatorial field at the same distance, and points along p; equatorial points opposite to p

Common mistakes

Dipole field falls as 1/r³, faster than a point charge's 1/r² — the two charges nearly cancel at large distance

Read the concept →

τ = pE sinθ; U = −pE cosθ

Torque and energy of a dipole in a uniform field

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
θ	angle between p and E	rad (or °)	[M⁰L⁰T⁰]
E	external uniform field	N/C	[MLT⁻³A⁻¹]

Valid when

Uniform field → net force zero, only torque
Stable equilibrium at θ = 0 (p along E), unstable at θ = 180°

Common mistakes

In a NON-uniform field a dipole also feels a net FORCE, not just torque

Read the concept →

Worth remembering

Work to rotate a dipole from θ₁ to θ₂: W = pE(cosθ₁ − cosθ₂); flipping from aligned to anti-aligned costs 2pE

Gauss's Law & Field of Symmetric Bodies

Gauss's law turns hard integrals into one-line answers whenever symmetry lets you choose the right closed surface.

Φ = ∮ E·dA = q(enc)/ε₀

Gauss's law

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
Φ	electric flux through the closed surface	N·m²/C (= V·m)	[ML³T⁻³A⁻¹]
q(enc)	net charge enclosed	C	[AT]

Valid when

Flux depends ONLY on enclosed charge — external charges contribute zero NET flux
The field E in the integral is due to ALL charges, inside and out

Common mistakes

A charge at the centre of a cube gives flux q/ε₀ total, so q/6ε₀ through each face — but only by symmetry; a charge at a corner needs q/8 enclosed

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Outside: E = kq/r²; On surface: kq/R²; Inside conductor: 0

Field of a charged sphere

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Gauss's law with a concentric spherical surface

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
R	radius of the sphere	m	[L]
r	distance from centre	m	[L]

Valid when

Outside: behaves as if all charge were at the centre
Solid NON-conductor (uniform charge): inside E = kqr/R³ (grows linearly from centre)

Common mistakes

Conductor vs uniformly-charged insulator differ INSIDE: conductor has E = 0, insulator has E ∝ r

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Line: E = λ/(2πε₀r); Sheet: E = σ/(2ε₀); Conductor surface: σ/ε₀

Field of a line charge and a sheet

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
λ	linear charge density	C/m	[ATL⁻¹]
σ	surface charge density	C/m²	[ATL⁻²]

Valid when

Infinite line: E ∝ 1/r; infinite sheet: E is UNIFORM (independent of distance)
Field just outside a charged conductor is σ/ε₀, twice that of an isolated sheet

Common mistakes

Between two oppositely charged parallel sheets E = σ/ε₀; outside them the field is zero

Read the concept →

Worth remembering

Flux through an open surface uses the same E·dA, but only a CLOSED surface gives q(enc)/ε₀

NEET asks flux-through-a-face and sphere-field-region questions; JEE Advanced hides Gauss inside cavity and superposition problems (charge in an off-centre cavity).

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