Moving Charges and MagnetismNEET Physics · Class 12 · NCERT Chapter 4

Introduction

A static electric charge produces an electric field. A moving electric charge (a current) produces a magnetic field. This chapter covers the magnetic force on moving charges and currents, the field they create, and a few important applications. Expect 1 to 2 NEET questions; favourites are Lorentz force, motion in B field, fields of straight wires and circular coils, and torque on current loops.

Lorentz force

A particle of charge q moving with velocity v in fields E and B feels a total force:

With only B, the magnetic part is $\vec{F}=q\vec{v}\times \vec{B}$, magnitude $qvB\sin \theta$, direction perpendicular to both v and B (right-hand rule). This force does no work because it is always perpendicular to v.

Magnetic force on a moving charge: F = q v B sin theta. Maximum when v is perpendicular to B; zero when parallel.

Force on a current-carrying wire

For a wire carrying current I in a field B, force on a small element dL:

For a straight wire of length L: $F=ILB\sin \theta$.

Charged particle in a uniform magnetic field

With v perpendicular to B, the magnetic force provides centripetal force:

Time period:

Period does NOT depend on speed. Faster particles trace larger circles in the same time.

With v perpendicular to B, the magnetic force becomes the centripetal force, so the particle moves in a circle. Radius depends on speed; period does not.

Cyclotron

A device for accelerating positive ions to high energies. Two D-shaped chambers (dees) sit in a perpendicular magnetic field B. An alternating voltage at the cyclotron frequency $f=qB/(2\pi m)$ accelerates the ion each time it crosses the gap. The ion spirals outward, gaining energy on each crossing.

Maximum kinetic energy: $K_{\max }=\frac{q^2{B}^2{R}^2}{2m}$, where R is the radius of the outermost orbit. Cyclotron does NOT work for electrons or relativistic ions, where T grows with speed.

Biot-Savart law

The magnetic field due to a current element $Id\vec{L}$ at point P at displacement r:

Constant ${\mu }_0=4\pi \times 10^{-7}\text{T m/A}$. Direction perpendicular to both dL and r.

Field of an infinite straight wire

At perpendicular distance r from a long straight wire carrying I:

Direction is given by the right-hand rule: curl fingers around the wire in the direction of current, thumb points along the field.

Magnetic field at perpendicular distance r from a long straight wire. Direction by the right-hand rule (curl fingers around the wire in the direction of current).

Field at the centre of a circular coil

For a single turn of radius R carrying current I, field at the centre:

For N turns: multiply by N. On the axis at distance x from centre:

Field at the centre and on the axis of a circular coil with N turns. At the centre, x = 0 and the formula simplifies.

Solenoid and toroid

A long solenoid is a tightly-wound helix carrying current I. Inside a long ideal solenoid:

Outside: zero. At the END: half the central value, ${\mu }_{0}nI/2$. A toroid is a solenoid bent into a doughnut shape; field inside the body is ${\mu }_{0}NI/(2\pi r)$, where r is the distance from the centre.

Inside a long ideal solenoid, the field is uniform along the axis and equal to mu_0 n I. At the end of a finite solenoid, B is half the value at the middle.

Ampere's circuital law

Line integral of B around any closed loop equals mu_0 times the enclosed current:

Like Gauss's law in electrostatics, Ampere's law is exact, but most useful when the geometry is symmetric enough to pull B out of the integral.

Force between parallel current-carrying wires

Two parallel wires carrying I_1 and I_2 separated by d:

Currents in the same direction: wires attract. Opposite directions: they repel.

Two parallel wires carrying currents. Same direction: they attract. Opposite directions: they repel.

Torque on a current loop in a magnetic field

A coil of N turns and area A carrying current I has a magnetic dipole moment:

In a uniform field B, the torque is:

Net force is zero, but the torque tries to align m with B.

Torque on a current loop in a uniform magnetic field. Used in galvanometers and electric motors. Torque tries to align m with B.

Moving coil galvanometer

A coil suspended in a radial magnetic field (provided by a soft iron core between the poles of a magnet). When current flows, torque $NIAB$ turns the coil. A spring provides restoring torque c θ. At equilibrium $NIAB=c\theta$, so:

Adding a low-resistance shunt makes it an ammeter; adding a high series resistance makes it a voltmeter.

Worked NEET problems

Summary cheat sheet

• Lorentz: $\vec{F}=q\vec{v}\times \vec{B}$, $F=qvB\sin \theta$.
• Wire force: $\vec{F}=I\vec{L}\times \vec{B}$.
• Circular motion: $r=mv/(qB)$, $T=2\pi m/(qB)$.
• Cyclotron: $f=qB/(2\pi m)$, $K_{\max }=q^2{B}^2{R}^2/(2m)$.
• Biot-Savart: $dB=({\mu }_0/4\pi )IdL\sin \theta /r^2$.
• Straight wire: $B={\mu }_{0}I/(2\pi r)$.
• Coil centre: $B={\mu }_{0}NI/(2R)$.
• Solenoid: $B={\mu }_{0}nI$ inside.
• Ampere's law: $\oint B\cdot dl={\mu }_0{I}_{\text{enc}}$.
• Parallel wires: $F/L={\mu }_0{I}_1{I}_2/(2\pi d)$.
• Magnetic moment: $m=NIA$, $\tau =mB\sin \theta$.

Next: try the interactive widgets for Lorentz force, Biot-Savart and torque on a loop, or work through the 32 NEET PYQs with full solutions. To time yourself, take the free 10-question mock test.