Maharashtra State Board · Class 12 · Physics · Chapter 10

Magnetic Fields due to Electric Current — Formula Sheet

Board Formulas

12 formulas

1.Biot–Savart Law★
$d B = \frac{μ _{0}}{4 π} \frac{I d l \times r ^}{r ^{2}}$
$μ_{0}$ : Permeability of free space = 4π×10⁻⁷ T·m/A · $I d l$ : Current element (A·m) · $r$ : Distance from element to field point (m)
Field due to a small current element I dl at position r from the element. Direction by right-hand rule.
2.Field at Centre of a Circular Loop★
$B = \frac{μ _{0} I}{2 R}$
For a circular loop of radius R carrying current I. Axis of loop is normal to plane.
3.Field on Axis of a Circular Loop★
$B = \frac{μ _{0} I R ^{2}}{2 ( R ^{2} + x ^{2} ) ^{3/2}}$
x is distance from centre along axis. Reduces to μ₀I/(2R) at x = 0.
4.Ampere's Circuital Law★
$\oint B \cdot d l = μ_{0} I_{e n c}$
Line integral of B around a closed Amperian loop = μ₀ × current enclosed.
5.Field due to a Long Straight Wire
$B = \frac{μ _{0} I}{2 π r}$
By Ampere's law with a circular Amperian loop of radius r around the wire.
6.Field of a Solenoid
$B = μ_{0} n I$
n = number of turns per unit length. Field is uniform inside a long solenoid.
7.Force on a Moving Charge (Lorentz Force)
$F = q (v \times B)$
Perpendicular to both v and B. No work is done on the charge — speed does not change.
8.Force on a Current-Carrying Conductor★
$F = I (L \times B), F = B I L sin θ$
θ is angle between L and B. Direction by Fleming's left-hand rule.
9.Cyclotron Frequency★
$f_{c} = \frac{q B}{2 π m}$
Independent of speed and radius — enables resonance in the cyclotron.
10.Maximum Kinetic Energy in Cyclotron
$K_{ma x} = \frac{q ^{2} B ^{2} R ^{2}}{2 m}$
R = maximum radius (dee radius). Sets a limit on ion energy.
11.Magnetic Moment of Current Loop
$m = I A = N I A$
N = number of turns. Direction from right-hand rule.
12.Gyromagnetic Ratio (orbital)★
$γ = \frac{m _{L}}{L} = \frac{e}{2 m _{e}}$
Ratio of magnetic moment to angular momentum of the orbital electron. Maharashtra-specific.