Formula sheet: Moving Charges and Magnetism
Moving Charges and Magnetism14 min readLesson 1 of 6
1.1 Force and motion
A charge moving in a magnetic field feels a force that is perpendicular to both the velocity and the field. The force is zero when the velocity is parallel to the field, and it never does work on the charge because it is always perpendicular to the motion. The field therefore bends the path but cannot change the speed.
For a charge moving perpendicular to a uniform field, the path becomes a circle. The magnetic force supplies the centripetal force, which balances the requirement for circular motion and fixes the radius and the time period. A charged particle moving at an angle to the field traces a helix.
The force on a current carrying conductor is the sum of the forces on its individual moving charges. A straight wire in a uniform field feels a force that is the cross product of the current direction and the field.
1.2 Fields produced by currents
A current element produces a magnetic field that falls as the square of the distance and depends on the angle between the element and the line joining it to the point. This is the Biot-Savart law. Integrating it over a distribution of current gives the field of a straight wire, a circular loop or a solenoid.
For a long straight wire the field circles the wire and falls as one over the distance from it. At the centre of a circular loop the field points along the axis, and on the axis it falls off with distance. Inside a long solenoid the field is nearly uniform and parallel to the axis, and outside it is very weak.
Ampere's law gives a quicker route when the symmetry is high. The line integral of the field around a closed loop equals the current enclosed times the permeability of free space. Use it for a long wire, a long solenoid and a toroid.
1.3 Instruments
A current loop behaves like a magnetic dipole. Its magnetic moment is the current times the area, directed along the normal to the loop by the right hand rule. In a uniform field the loop feels no net force, but it does feel a torque that tends to align the moment with the field.
The moving coil galvanometer uses this torque. A coil suspended in a radial magnetic field turns until the spring torque balances the magnetic torque, and the deflection is proportional to the current. With a small shunt resistance in parallel the galvanometer becomes an ammeter; with a large resistance in series it becomes a voltmeter.
| Quantity | Expression | Note |
|---|---|---|
| Magnetic moment of a loop | m = N I A | unit A m^2, direction by right hand rule |
| Galvanometer deflection | phi = (N A B / k) I | k is the torsional constant |
| Ammeter shunt | S = I_g G / (I - I_g) | small resistance in parallel |
| Voltmeter series resistance | R = V / I_g - G | large resistance in series |
| Current sensitivity | phi / I = N A B / k | increases with N, A and B |
| Voltage sensitivity | phi / V = N A B / (k G) | independent of the coil turns for fixed geometry |
| Permeability | mu0 = 4 pi × 10^-7 T m A^-1 | free space value |
Exam
Direction questions need the right hand rule stated in words. Write the vectors, apply the cross product, then state the direction. For circular motion questions, show that the magnetic force is the centripetal force before solving for the radius.