Class 12 · Chapter 4

Moving Charges & Magnetism

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Moving Charges & Magnetism Formula Sheet

9 formulas across 3 topics in Moving Charges & Magnetism.

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Updated 2026-07-05 · v1.0.0

Magnetic Force on Charges & Currents

A magnetic field does no work — it only bends paths. The Lorentz force is always perpendicular to velocity, turning charges into circles.

F = qvB sinθ; F = q(v × B)

Lorentz (magnetic) force on a charge

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
F	magnetic force	N	[MLT⁻²]
q	charge	C	[AT]
v	velocity of the charge	m/s	[LT⁻¹]
B	magnetic field	T (tesla)	[MT⁻²A⁻¹]
θ	angle between v and B	rad (or °)	[M⁰L⁰T⁰]

Valid when

Force is perpendicular to BOTH v and B — magnetic force does zero work, so speed and KE never change
Zero force when v is parallel to B (θ = 0)

Common mistakes

A magnetic field cannot change a particle's speed or kinetic energy — only its direction

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r = mv/(qB); T = 2πm/(qB); f = qB/(2πm)

Charged particle in a magnetic field

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Magnetic force supplies the centripetal force: qvB = mv²/r

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
r	radius of the circular path	m	[L]
T	time period of revolution	s	[T]

Valid when

Period is INDEPENDENT of speed and radius — the cyclotron principle
Velocity component along B is unaffected → helical path when v has a parallel component

Common mistakes

Faster particles trace BIGGER circles but take the SAME time per revolution

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F = BIL sinθ; F = I(L × B)

Force on a current-carrying wire

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
I	current	A	[A]
L	length of the wire in the field	m	[L]
θ	angle between the wire and B	rad (or °)	[M⁰L⁰T⁰]

Valid when

Force per unit length between two parallel wires: F/L = μ₀I₁I₂/(2πd) — attract if currents are parallel

Common mistakes

A closed current loop in a UNIFORM field feels zero NET force (but a torque) — the side forces cancel

Read the concept →

Worth remembering

Velocity selector: E and B crossed so qE = qvB, passing only v = E/B undeflected
Cyclotron frequency qB/2πm sets the RF driving frequency — independent of the ion's energy

Magnetic Field of Currents

Biot-Savart is to magnetism what Coulomb is to electricity — the elementary law you integrate. Ampère's law is its Gauss-style shortcut for symmetric cases.

dB = (μ₀/4π) I dl sinθ / r²

Biot-Savart law

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
μ₀	permeability of free spaceμ₀ = 4π × 10⁻⁷ T·m/A	T·m/A	[MLT⁻²A⁻²]
dl	current element length	m	[L]
θ	angle between dl and the position vector r	rad (or °)	[M⁰L⁰T⁰]

Valid when

dB is perpendicular to both dl and r (right-hand rule)

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B = μ₀I/(2πr) (infinite wire); B = (μ₀I/4πr)(sinθ₁ + sinθ₂) (finite)

Field of a straight current-carrying wire

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Integrating Biot-Savart along the wire

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
r	perpendicular distance from the wire	m	[L]

Valid when

Field circles the wire (right-hand grip rule); B ∝ 1/r

Common mistakes

The infinite-wire formula μ₀I/2πr is a special case of the finite formula with θ₁ = θ₂ = 90°

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Centre of loop: B = μ₀NI/(2R); On axis: B = μ₀NIR²/[2(R² + x²)^(3/2)]

Field of a circular current loop

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
N	number of turns	dimensionless	[M⁰L⁰T⁰]
R	loop radius	m	[L]
x	distance along the axis from the centre	m	[L]

Valid when

A current loop behaves like a magnetic dipole of moment m = NIA

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Solenoid: B = μ₀nI; Toroid: B = μ₀NI/(2πr)

Field of a solenoid and toroid

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Ampère's law ∮B·dl = μ₀I(enc) applied to a rectangular / circular loop

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
n	turns per unit length	m⁻¹	[L⁻¹]
N	total number of turns (toroid)	dimensionless	[M⁰L⁰T⁰]

Valid when

Field inside a long solenoid is UNIFORM and independent of position; outside ≈ 0
Field at the END of a solenoid is half the interior value: μ₀nI/2

Common mistakes

Solenoid field depends on turns per unit LENGTH (n), not total turns N

Read the concept →

Worth remembering

Ampère's law ∮B·dl = μ₀I(enc) works only where symmetry makes B constant along the chosen loop

Torque on Loops & the Galvanometer

A current loop in a field is a magnetic dipole; the torque it feels is the working principle of every moving-coil meter.

τ = NIAB sinθ = mB sinθ; m = NIA

Torque on a current loop; magnetic moment

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
m	magnetic dipole moment	A·m²	[AL²]
A	area of the loop	m²	[L²]
θ	angle between the loop's normal (m) and B	rad (or °)	[M⁰L⁰T⁰]

Valid when

Maximum torque when the loop PLANE is parallel to B (θ = 90°); zero when the plane is perpendicular

Common mistakes

θ is measured from the NORMAL to the loop, not from the loop plane — the two differ by 90°

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Ammeter: shunt S = I(g)G/(I − I(g)); Voltmeter: series R = V/I(g) − G

Galvanometer to ammeter / voltmeter

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
G	galvanometer resistance	Ω	[ML²T⁻³A⁻²]
I(g)	full-scale deflection current	A	[A]
S	shunt resistance (small, parallel)	Ω	[ML²T⁻³A⁻²]

Valid when

Ammeter: low resistance in SERIES with the circuit, needs a small PARALLEL shunt — ideal ammeter has zero resistance
Voltmeter: high resistance in PARALLEL with the element, needs a large SERIES resistor — ideal voltmeter has infinite resistance

Common mistakes

Swapping shunt (parallel, ammeter) and series-resistor (voltmeter) roles is the classic instrument-conversion error

Read the concept →

Worth remembering

Moving-coil galvanometer sensitivity rises with N, A and B, and falls with the spring constant k

NEET repeats radius/period, solenoid field and galvanometer-conversion; JEE Advanced favours helical motion, crossed fields, and Ampère's law on non-obvious geometries.

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