CBSE · Class 12 · Physics · Chapter 4

Moving Charges and Magnetism — Formula Sheet

Board Formulas

13 formulas

1.Lorentz Force★
$F = q (E + v \times B)$
$q$ : Charge (C) · $v$ : Velocity of charge (m/s) · $B$ : Magnetic field (T)
Total electromagnetic force on a charge. Magnetic part is perpendicular to v — does no work.
2.Biot–Savart Law★
$d B = \frac{μ _{0}}{4 π} \frac{I d l \times r ^}{r ^{2}}$
Field due to a small current element. Analogue of Coulomb's law for magnetism.
3.Field due to Long Straight Wire
$B = \frac{μ _{0} I}{2 π r}$
Field lines are concentric circles around the wire; right-hand thumb along I.
4.Field at Centre of Circular Loop★
$B = \frac{μ _{0} I}{2 R}$
For N turns, multiply by N: B = μ_0 NI/(2R). Direction perpendicular to plane of loop.
5.Field on Axis of Circular Loop
$B = \frac{μ _{0} I R ^{2}}{2 ( R ^{2} + x ^{2} ) ^{3/2}}$
At the centre (x = 0) reduces to μ_0 I/(2R). Falls as 1/x³ far away.
6.Ampere's Circuital Law★
$\oint B \cdot d l = μ_{0} I_{e n c}$
Line integral of B around a closed path = μ_0 × enclosed current. Only useful with symmetry.
7.Field Inside a Solenoid
$B = μ_{0} n I$
n = turns per unit length (m⁻¹). Field is uniform inside, ≈ zero outside (ideal case).
8.Field Inside a Toroid
$B = \frac{μ _{0} N I}{2 π r}$
N = total number of turns. Field exists only within the toroidal core.
9.Force on Current-Carrying Conductor
$F = I L \times B, F = B I L sin θ$
Maximum when L is perpendicular to B (θ = 90°). Zero when parallel.
10.Force Between Two Parallel Wires
$\frac{F}{L} = \frac{μ _{0} I _{1} I _{2}}{2 π d}$
Attractive if currents parallel, repulsive if antiparallel. Defines the ampere.
11.Torque on a Current Loop
$τ = m \times B, τ = N I A B sin θ$
Magnetic dipole moment m = NIA. Torque is zero when m is aligned with B.
12.Cyclotron Frequency
$f = \frac{q B}{2 π m}, T = \frac{2 π m}{q B}$
Independent of speed and radius (in non-relativistic limit) — this is why cyclotrons work.
13.Radius of Circular Motion
$r = \frac{m v}{q B}$
Faster or heavier particles orbit in larger circles. Doubling B halves the radius.