Class 12 Physics · Chapter 4 NotesMoving Charges and Magnetism

A stationary charge produces an electric field, but a moving charge or current also produces a magnetic field. The total force on a charge q moving with velocity v in both electric field E and magnetic field B is called the Lorentz force: F = q(E + v × B). The magnetic part q(v × B) is always perpendicular to both v and B, so it does no work and cannot change the speed of the particle. The force is zero when the charge is at rest or when v is parallel or antiparallel to B. For a straight conductor of length l carrying current I in a uniform external field B, the force is F = I l × B, where the direction of l is along the current.

• Lorentz forceF = q(E + v × B)
• Magnetic force magnitudeF = q v B sin θ
• Force on a conductorF = I l × B
• Unit of magnetic fieldtesla (T); 1 gauss = 10⁻⁴ T

When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force acts as a centripetal force and the particle moves in a circle. Equating m v²/r to q v B gives the radius r = m v / (q B). The angular frequency is ω = q B / m, and the cyclotron frequency is ν = q B / (2π m), which is independent of speed and radius. If the velocity has a component along B, that component remains unchanged while the perpendicular component causes circular motion, resulting in a helical path. The distance moved along the field in one revolution is called the pitch, p = v∥ T = 2π m v∥ / (q B).

• Radius of circular pathr = m v / (q B)
• Cyclotron frequencyν = q B / (2π m)
• Pitch of helixp = 2π m v∥ / (q B)
• Magnetic force does no work; speed remains constant

The Biot-Savart law gives the magnetic field dB produced by a small current element I dl at a distance r: dB = (μ₀/4π) (I dl × r) / r³. Its magnitude is dB = (μ₀/4π) (I dl sin θ) / r², where θ is the angle between dl and r. The constant μ₀ is the permeability of free space, with μ₀/4π = 10⁻⁷ T m A⁻¹. Unlike the electric field, the magnetic field is perpendicular to the plane containing dl and r. For a circular loop of radius R carrying current I, the field on the axis at distance x from the centre is B = μ₀ I R² / [2 (x² + R²)^(3/2)]. At the centre (x = 0), B = μ₀ I / (2R).

• Biot-Savart lawdB = (μ₀/4π) (I dl × r) / r³
• μ₀ = 4π × 10⁻⁷ T m A⁻¹
• Field at centre of circular loopB = μ₀ I / (2R)
• Field on axisB = μ₀ I R² / [2 (x² + R²)^(3/2)]

Ampere's circuital law states that the line integral of B around a closed loop equals μ₀ times the total current enclosed: ∮ B · dl = μ₀ I. For symmetric cases, if B is tangential and constant along a path of length L, then B L = μ₀ Iₑ. Applying this to a long straight wire gives B = μ₀ I / (2π r) outside the wire. Inside a wire of radius a with uniform current density, B = (μ₀ I / 2π a²) r, so B ∝ r for r < a and B ∝ 1/r for r > a. For a long solenoid with n turns per unit length, the field inside is uniform: B = μ₀ n I.

• Ampere's law∮ B · dl = μ₀ I
• Long straight wireB = μ₀ I / (2π r)
• Inside wire (r < a)B = (μ₀ I / 2π a²) r
• Long solenoidB = μ₀ n I

Two long parallel conductors carrying currents Iₐ and I_b separated by distance d exert magnetic forces on each other. Conductor a produces a field Bₐ = μ₀ Iₐ / (2π d) at the location of conductor b. The force on a length L of conductor b is F_ba = μ₀ Iₐ I_b L / (2π d). The force per unit length is f = μ₀ Iₐ I_b / (2π d). Currents flowing in the same direction attract each other, while antiparallel currents repel. This is opposite to the behaviour of like electric charges. This force definition is used to define the ampere, the SI unit of current.

• Force per unit lengthf = μ₀ Iₐ I_b / (2π d)
• Parallel currents attract; antiparallel currents repel
• Ampere defined using force between two long parallel conductors
• Newton's third law holds for steady currents

A rectangular loop of area A carrying current I in a uniform magnetic field B experiences no net force but a torque. The torque is τ = I A B sin θ, where θ is the angle between the normal to the loop and B. Defining the magnetic moment m = I A (direction given by the right-hand thumb rule), the torque is τ = m × B. For N turns, m = N I A. The torque is zero when m is parallel or antiparallel to B; parallel is stable equilibrium, antiparallel is unstable. A current loop behaves like a magnetic dipole at large distances, with field B = (μ₀/4π)(2m/x³) on the axis.

• Torque on loopτ = m × B
• Magnetic momentm = N I A
• Unit of mA m²
• Stable equilibrium when m is parallel to B

A moving coil galvanometer detects and measures small currents. It consists of a coil of N turns suspended in a uniform radial magnetic field produced by a cylindrical soft iron core. When current I flows, the magnetic torque N I A B is balanced by the spring's restoring torque k φ, giving φ = (N A B / k) I. The current sensitivity is φ/I = N A B / k. To convert it into an ammeter, a small shunt resistance r_s is connected in parallel so most current bypasses the galvanometer. To convert it into a voltmeter, a large resistance R is connected in series to draw very little current.

• Equilibriumk φ = N I A B
• Current sensitivityφ/I = N A B / k
• Ammetersmall shunt resistance in parallel
• Voltmeterlarge resistance in series