Calculating Magnetic Fields and Forces

Magnetostatics

Our study of the electric field built on our previous understanding of electric forces, which was ultimately based on Coulomb's law for the electric force between two point charges. Since magnetism is ultimately an interac tion between currents, i.e. between moving charges, it is reasonable to wish for a magnetic analog of Coulomb's law, an equation that would tell us the magnetic force between any two moving point charges.

Such a law, unfortunately, does not exist. Coulomb's law describes the special case of electrostatics: if a set of charges is sitting around and not moving, it tells us the interactions among them. Coulomb's law fails if the charges are in motion, since it does not incorporate any allowance for the time delay in the outward propagation of a change in the locations of the charges.

A pair of moving point charges will certainly exert magnetic forces on one another, but their magnetic fields are like the v-shaped bow waves left by boats. Each point charge experiences a magnetic field that originated from the other charge when it was at some previous position. There is no way to construct a force law that tells us the force between them based only on their current positions in space.

There is, however, a science of magnetostatics that covers a great many important cases. Magnetostatics describes magnetic forces among currents in the special case where the currents are steady and continuous, leading to magnetic fields throughout space that do not change over time.

If we cannot build a magnetostatics from a force law for point charges, then where do we start? It can be done, but the level of mathematics required (vector calculus) is inappropriate for this course. Luckily there is an alternative that is more within our reach. Physicists of generations past have used the fancy math to derive simple equations for the fields created by various static current distributions, such as a coil of wire, a circular loop, or a straight wire. Virtually all practical situations can be treated either directly using these equations or by doing vector addition, e.g. for a case like the field of two circular loops whose fields add onto one another.

The figure shows the equations for some of the more commonly encountered configurations, with illustrations of their field patterns. Do not memorize the equations! The symbol μ_o is an abbreviation for the constant 4πx10 ^-7 T.m/A. It is the magnetic counterpart of the Coulomb force constant k. The Coulomb constant tells us how much electric field is produced by a given amount of charge, while μ_o relates currents to magnetic fields. Unlike k, μ_o has a definite numerical value because of the design of the metric system.

Field created by a long, straight wire carrying current I:

Here r is the distance from the center of the wire. The field vectors trace circles in planes perpendicular to the wire, going clockwise when viewed from along the direction of the current.

Field created by a single circular loop of current:

The field vectors form a dipole-like pattern, coming through the loop and back around on the outside. Each oval path traced out by the field vectors appears clockwise if viewed from along the direction the current is going when it punches through it. There is no simple equation for the field at an arbitrary point in space, but for a point lying along the central axis perpendicular to the loop, the field is

where b is the radius of the loop and z is the distance of the point from the plane of the loop.

Field created by a solenoid (cylindrical coil):

The field pattern is similar to that of a single loop, but for a long solenoid the paths of the field vectors become very straight on the inside of the coil and on the outside immediately next to the coil. For a sufficiently long solenoid, the interior field also becomes very nearly uniform, with a magnitude of

B = μ_oIN/$ ,

where N is the number of turns of wire and $ is the length of the solenoid. The field near the mouths or outside the coil is not constant, and is more difficult to calculate. For a long solenoid, the exterior field is much smaller than the interior field.

Force on a charge moving through a magnetic field

We now know how to calculate magnetic fields in some typical situa tions, but one might also like to be able to calculate magnetic forces, such as the force of a solenoid on a moving charged particle, or the force between two parallel current-carrying wires.

We will restrict ourselves to the case of the force on a charged particle moving through a magnetic field, which allows us to calculate the force between two objects when one is a moving charged particle and the other is one whose magnetic field we know how to find. An example is the use of solenoids inside a TV tube to guide the electron beam as it paints a picture.

Magnetic forces cause a beam of electrons to move in a circle. The beam is created in a vacuum tube, in which a small amount of hydrogen gas has been left. A few of the electrons strike hydrogen molecules, creating light and letting us see the beam. A magnetic field is produced by passing a current (meter) through the circular coils of wire in front of and behind the tube. In the bottom figure, with the magnetic field turned on, the force perpendicular to the electrons' direction of motion causes them to move in a circle.

Experiments show that the magnetic force on a moving charged particle has a magnitude given by

|F| = q| ||B|sin θ

where v is the velocity vector of the particle, and θ is the angle between the v and B vectors. Unlike electric and gravitational forces, magnetic forces do not lie along the same line as the field vector. The force is always perpendicular to both v and B. Given two vectors, there is only one line perpendicular to both of them, so the force vector points in one of the two possible directions along this line. For a positively charged particle, the direction of the force vector is the one such that if you sight along it, the B vector is clockwise from the v vector; for a negatively charged particle the direction of the force is reversed. Note that since the force is perpendicular to the particle's motion, the magnetic field never does work on it.

Hallucinations during an MRI scan