What to Memorise — Moving Charges & Magnetism

Group 01

Formulas that must be automatic

If you have to derive it in the exam, you have already lost the question

Forces
Lorentz force (both fields)	F = q(E + v × B)
Magnetic part alone	F = qvB sin θ
On a straight wire	F = BIL sin θ · F = I l × B
Between parallel wires, per metre	f = μ₀ I₁ I₂ / 2πd

Motion of a charged particle
Radius of the circle	r = m v⊥ / qB
Time for one lap	T = 2πm / qB
Frequency and angular frequency	ν = qB / 2πm · ω = qB / m
Pitch of the helix	p = v∥ × 2πm / qB

Fields produced by currents
Biot–Savart, one element	dB = (μ₀/4π) I dl sin θ / r²
Infinite straight wire	B = μ₀I / 2πr
Finite straight wire	B = (μ₀I / 4πd)(sin θ₁ + sin θ₂)
Centre of a circular loop	B = μ₀NI / 2R
On the axis of a loop	B = μ₀IR² / 2(x² + R²)^{3/2}
Arc subtending θ radians	B = μ₀Iθ / 4πR
Ampère's circuital law	∮ B·dl = μ₀ I(enclosed)
Inside a thick wire (r < a)	B = μ₀Ir / 2πa²
Inside a long solenoid	B = μ₀nI
Inside a toroid	B = μ₀NI / 2πr

Loops, meters and machines
Magnetic dipole moment	m = NIA
Torque on a loop	τ = m × B = NIAB sin θ
Dipole field, on the axis	B = μ₀ 2m / 4πx³
Dipole field, sideways	B = μ₀ m / 4πx³
Galvanometer deflection	φ = (NAB / k) I
Current sensitivity	φ/I = NAB / k
Voltage sensitivity	φ/V = NAB / kR
Shunt to make an ammeter	S = I_g G / (I − I_g)
Series R to make a voltmeter	R = V / I_g − G
Velocity selector	v = E / B
Cyclotron frequency	f = qB / 2πm
Cyclotron maximum energy	K = q²B²R² / 2m

Group 02

Constants, with no hesitation

The two shortcut forms of μ₀ save more time than anything else on this page

Numbers
Permeability of free space	μ₀ = 4π × 10⁻⁷ T·m/A
Shortcut for Biot–Savart	μ₀/4π = 10⁻⁷
Shortcut for straight and parallel wires	μ₀/2π = 2 × 10⁻⁷
Elementary charge	e = 1.6 × 10⁻¹⁹ C
Electron mass	mₑ = 9.1 × 10⁻³¹ kg
Proton mass	mₚ = 1.67 × 10⁻²⁷ kg
Gauss to tesla	1 gauss = 10⁻⁴ T
Earth's field	≈ 3.6 × 10⁻⁵ T (horizontal part ≈ 3 × 10⁻⁵ T)
Electron-volt	1 eV = 1.6 × 10⁻¹⁹ J
The hidden gem	μ₀ε₀ = 1 / c²

Group 03

Direction rules

Right hand only — the left hand gives every answer backwards

v × B — fingers along v, curl toward B, thumb gives F.
Straight wire — grip it, thumb along the current, fingers curl the way B goes.
Loop or solenoid — curl fingers along the current, thumb gives B and m.
Unit-vector ring — î → ĵ → k̂ → î. Forward gives plus, backward gives minus.
Page notation — ⊙ out of the page · ⊗ into the page.
Parallel currents attract; antiparallel repel. Backwards from charges.
Order matters — A × B = − B × A.

Why this mattersMost sign errors in this chapter are one of these seven rules applied backwards. They are not really seven separate things to learn — they are the same right hand used four different ways.

Group 04

The zeros

NEET loves these because the answer needs no arithmetic at all

A stationary charge feels no magnetic force, however strong the field.
A charge moving parallel or antiparallel to B feels no force.
Work done by a magnetic force is always zero — speed never changes.
Net force on a closed loop in a uniform field is zero. Only torque survives.
Straight leads pointing at the centre of an arc contribute nothing.
A current element makes no field along its own direction.
On the axis of a thick wire (r = 0) the field is zero.
In a toroid's central hole, and outside the toroid, the field is zero.
Outside an ideal long solenoid the field is zero.

Group 05

The “independent of” facts

Straight recall questions — each one is a mark on offer

What does not depend on what
Period T and frequency ν are independent of…	speed and radius → this is why a cyclotron works
Solenoid B = μ₀nI is independent of…	the radius, and of where inside you stand
Velocity selector v = E/B is independent of…	charge and mass → works on any particle at all
Cyclotron maximum energy is independent of…	the accelerating voltage (that only sets the number of laps)
Torque on a loop is independent of…	the loop's shape — only its area matters

Group 06

Standard results and ratios

Short answers that appear again and again in past papers

Worth knowing outright
Semicircle, and quarter circle	½ and ¼ of the full-loop field
Loop axis at x = R	1 / 2√2 = 0.354 of the centre value
Same wire rebent into n turns	B ∝ n² (n from turns, n from halved radius)
Old definition of the ampere	2 × 10⁻⁷ N/m between 1 A wires 1 m apart
Fall-off with distance	1/r straight wire · 1/r² one element · 1/r³ dipole · B ∝ r inside a thick wire
Ideal meters	ammeter 0 Ω in series · voltmeter ∞ Ω in parallel

Group 07

Units and dimensions

Asked directly most years, and they take thirty seconds to learn

Dimensional formulas
Magnetic field B	tesla (T) · [M T⁻² A⁻¹]
Magnetic moment m	A·m² or J/T · [L² A]
Permeability μ₀	T·m/A · [M L T⁻² A⁻²]
Torsion constant k	N·m/rad · [M L² T⁻²]

Group 08

Names, years and claims to fame

Cheap marks, and they turn up in assertion–reason questions

Who did what
Hans Christian Oersted, 1820	compass needle deflects near a current — the first link between the two
André-Marie Ampère	circuital law; proposed that all magnetism is circulating currents
Biot and Savart	the law giving the field of one current element
Hendrik Lorentz	the combined force law; Nobel Prize 1902 for the Zeeman effect
James Clerk Maxwell, 1864	unified electricity and magnetism; showed light is an electromagnetic wave
Nikola Tesla · Carl Friedrich Gauss	the two units of magnetic field
Ernest Lawrence, 1932	built the first cyclotron

Group 09

What NOT to memorise

Understand these once; NEET asks for the result, never the working

The full Biot–Savart integration for the field on a loop's axis.
The Ampèrian rectangle derivation for the solenoid.
The couple-and-torque derivation for a rectangular loop.
The drift-velocity derivation of F = BIL from single charges.

The real splitRoughly seven-tenths of this chapter should be understood and re-derived on the spot. What is on this page is the other three-tenths — the part where deriving it costs you the question.