Magnetic FieldsMagnetic Field

Forces on a Charged ParticleMagnetic Field Lines

Crossed Fields and Hall EffectCirculating Charged ParticlesCyclotrons and Synchrotrons

Magnetic Force on a Current Carrying WireTorque on a Current LoopMagnetic Dipole Moment

An electric field produces an electric force. Similarly a magnetic field can produce a magnetic force. The magnetic field is denoted by the symbol, . One might expect that a magnetic charge produces the field much like an electric charge. Magnetic monopoles, as these are called are predicted by some theories, but their existence has not been confirmed.

One way to produce a magnetic field is to use moving electric charges to create an electromagnet. This is done in motors, telephones and computer disk drives as well as many other places. Magnetic fields are also produced by some minerals and ores because the molecules and atoms have an intrinsic magnetic field associated with them. These are permanent magnets. In most materials the magnetic fields on various atoms and molecules cancel out.

If magnetic monopoles exist, the magnetic field would equal the force on the particle divided by the magnetic monopole. Since the monopoles have not yet been detected, that definition can not be used. One can define the magnetic field to be directed along the zero force axis and the velocity is perpendicular to the field.

sinF

F qv F q vq v

The magnetic force acting on a charged particle moving with a velocity, v, through a magnetic field, , is always perpendicular to both the velocity and the magnetic field.

The SI unit for the magnetic field is the tesla (T), one newton per coulomb per metre per second or one newton per ampere per metre. Another unit still in use is the gauss. One tesla equals 10⁴ gauss.

Opposite poles attract and like poles repel one another.

Both electric fields and magnetic fields produce a force on a charged particle. When the two fields are perpendicular to each other they are called crossed fields. J. J. Thomson used a cathode ray tube with crossed fields to discover the electron. Using plates of length, L, which show a deflection, y, from the straight line path, one can determine the ratio of the mass to the charge on the particle moving through the cathode ray tube.

2 2 2

22 2

q L m Ly q q v v

mv q y

In 1879, Edwin Hall demonstrated that electrons moving through a copper wire can be deflected by a magnetic field. This deflection allows us to determine if the charge carriers are positive or negative. A potential difference exists across the width, d, of the conductor.

In order for a charged particle to move in a circle, there must be a centripetal force. That force can be a magnetic force.

If a charged particle enters the magnetic field at an angle with a component parallel and perpendicular to the magnetic field, the parallel component causes the particle to move in a helical path with a pitch equal to the component in parallel direction. The pitch is the distance between adjacent turns of the helix.

H d d

JV d e ev v n

Ane neA V ed

2 2 2 2

12

2

mv mv r mv mF q v r T

r q v v q q

q qff

T m m

Electrons and protons can be used to study atomic structure. High energy beams can be created by shooting them into a magnetic field.

A cyclotron consists of two hollow D shaped, copper containers, open along the straight edge. These dees as they are called, are part of an electrical oscillator that alternates the electric potential across the gap between the parts first in one direction then the other. A magnetic field is applied perpendicular to the dees. By choosing the correct frequency of the oscillator, the particle is given more energy as it moves from one dee to the other causing it to move faster and making the particle spiral outward. This works up to an energy of 50 MeV.

A synchrotron has an oscillator and magnetic field which are variable allowing the energies to reach higher values, even exceeding 1 TeV.

Because of the Hall Effect, if a current carrying wire is placed in a magnetic field it will feel a force which is dependent on the current, length of the wire and magnetic field.

d dd d

L Lq t F qv v L F L

v v

A loop of wire carrying a current in a magnetic field is subject to a torque if the plane of the loop is not perpendicular to the direction of the magnetic field. The direction of the plane of the loop is a vector which is perpendicular to the plane of the loop and points according to the right hand rule. One can measure the angle between the direction of the magnetic field and the vector defining the plane of the loop. The torque trying to align the vector of the loop with the magnetic field depends on the number of loops, the current, the area of the loop and the magnetic field.

The quantity inside the parentheses in the equation above is called the magnetic moment.

The magnetic potential energy that depends on the orientation to the magnetic field.

sinN A

sinN A N A

fiU W U U U