JEE Physics Moving Charges and Magnetism Notes Set 05

Moving Charges and Magnetism: Quick Revision

A moving charge or a current produces a magnetic field. The Biot-Savart law gives the field of a small current element: \( dB=\frac{\mu_0}{4\pi}\frac{I\,dl\sin\theta}{r^2} \), directed by the right-hand rule. Ampere's circuital law, \( \oint\vec B\cdot d\vec l=\mu_0I_{enclosed} \), is the quick tool when the situation is symmetric. A wire segment whose line passes through the point of observation contributes nothing there.

Standard Fields

Inside a solid current-carrying wire of radius a the field grows linearly, \( B=\frac{\mu_0Ir}{2\pi a^2} \), and outside it falls as \( \frac1r \). Far from a coil on its axis the field falls as \( \frac1{r^3} \). If a wire of fixed length is wound into N turns, the radius becomes 1/N times and the central field becomes \( N^2 \) times larger.

Motion of a Charge in Fields

The magnetic force is \( \vec F=q(\vec v\times\vec B) \); it is perpendicular to the velocity, does no work and cannot change the speed. If the velocity is perpendicular to B the path is a circle of radius \( r=\frac{mv}{qB}=\frac{\sqrt{2mK}}{qB} \), with time period \( T=\frac{2\pi m}{qB} \) independent of speed. If the velocity makes an angle \( \theta \) with B the path is a helix of radius \( \frac{mv\sin\theta}{qB} \) and pitch \( \frac{2\pi m}{qB}v\cos\theta \).

When both fields act, the Lorentz force is \( \vec F=q\vec E+q(\vec v\times\vec B) \). A particle moves undeflected when E, B and v are mutually perpendicular and \( v=\frac EB \) (velocity selector). For particles accelerated through the same potential V, the radius goes as \( \sqrt{\frac mq} \); for equal kinetic energy and equal charge it goes as \( \sqrt m \); for equal momentum it goes as \( \frac1q \). A cyclotron works because T does not depend on speed, and its maximum kinetic energy is \( \frac{q^2B^2R^2}{2m} \).

Forces on Currents and Loops

A straight conductor in a uniform field feels \( \vec F=I(\vec L\times\vec B) \), where \( \vec L \) is the vector from start to end, so a curved wire acts like a straight wire joining its ends. The net force on a closed loop in a uniform field is zero. Two long parallel wires exert a force per unit length \( \frac{\mu_0I_1I_2}{2\pi d} \): parallel currents attract and antiparallel currents repel.

A loop with magnetic moment \( \vec M=NI\vec A \) in a field feels torque \( \vec\tau=\vec M\times\vec B \) and has energy \( U=-\vec M\cdot\vec B \). A revolving charge gives a moment \( M=\frac12qvr \). A moving-coil galvanometer uses the relation NIAB = k times the deflection, and a tangent galvanometer uses \( B_{coil}=B_H\tan\theta \).