JEE Physics Moving Charges and Magnetism Notes Set 04

Force on Charges and Currents

A charge q moving with velocity \( \vec v \) in a magnetic field \( \vec B \) feels \( \vec F=q\,\vec v\times\vec B \). The force is perpendicular to both velocity and field, so it changes the direction of motion but never the speed or kinetic energy.
A straight wire of length l carrying current I experiences \( F=IlB\sin\theta \), where \( \theta \) is the angle between the wire and the field. The direction follows Fleming's left-hand rule.
If the particle enters at right angles to a uniform field it moves on a circle of radius \( r=\frac{mv}{qB} \) with period \( T=\frac{2\pi m}{qB} \), which does not depend on speed. When it is accelerated through a potential V first, \( r=\frac{1}{B}\sqrt{\frac{2mV}{q}} \).

Field Produced by Currents

Biot-Savart law: a current element \( Id\vec l \) produces \( dB=\frac{\mu_0}{4\pi}\frac{I\,dl\sin\theta}{r^2} \).
Long straight wire: \( B=\frac{\mu_0 I}{2\pi d} \). Finite wire: \( B=\frac{\mu_0 I}{4\pi d}(\sin\alpha+\sin\beta) \). A point on the line of a wire gets no field from it.
Centre of a circular coil of N turns: \( B=\frac{\mu_0NI}{2r} \). Arc subtending angle \( \theta \) at its centre: \( B=\frac{\mu_0I\theta}{4\pi r} \), so a semicircle gives \( \frac{\mu_0I}{4r} \). On the axis of a loop: \( B=\frac{\mu_0Ir^2}{2(r^2+x^2)^{3/2}} \).
Long solenoid: \( B=\mu_0nI \) inside, where n is the number of turns per unit length, and nearly zero outside. With a core of relative permeability \( \mu_r \) the field is \( \mu_r\mu_0nI \). Magnetic intensity \( H=nI \).

Forces Between Currents, Loops and Galvanometer

Two parallel wires carrying currents in the same direction attract and in opposite directions repel, with force per unit length \( \frac{\mu_0I_1I_2}{2\pi d} \).
A current loop of area A with N turns has magnetic moment \( m=NIA \). In a field it feels torque \( \tau=mB\sin\theta \) and has potential energy \( U=-mB\cos\theta \); turning it through 180° from the stable position needs work 2mB.
Moving coil galvanometer: the deflection obeys \( NIAB=k\phi \). Current sensitivity is \( \frac{\phi}{I}=\frac{NAB}{k} \) and voltage sensitivity is \( \frac{\phi}{V}=\frac{NAB}{kR} \). Increasing the number of turns raises the coil resistance in the same proportion, so voltage sensitivity does not improve.