Magnetic Field and Forces

The Magnetic Field and Magnetic Materials

Most people are familiar with magnetic phenomena and have an idea that magnetic materials are mostly metal. In fact, magnetism is one of the forces in nature that most easily manifests itself as one can generally "feel it". This explains the popularity of fridge magnets.
Most of us have done experiments with bar magnets as well and intuitively know that there is a north and south pole. Indeed, this is one of the fundamental properties of magnetic materials -there are two poles.
Like poles repel and opposite poles attract. This produces a pattern of field lines. This pattern can be revealed using a compass and a bar magnetic or by placing a bar magnet amongst iron filings.

The bipolar field lines associated with a bar magnetic should remind you of the electric dipole field discussed in an earlier unit. However, if a magnetic dipole were strictly like that of an electric dipole, it should be possible to separate the dipole into its individual components like this:

But we know from every day experience that this does not happen and instead we produce two magnets, each with and N and S pole.

First Visualization

Earth's Magnetic Field

Magnetism is a force that acts as a distance and can permeate many materials relatively unimpeded. A familiar example of this is the Earth's magnetic field. A compass can "read" the Earth's magnetic field without actually touching anything and it can make this reading at many different locations on the Earth.

Indeed the large scale force of the Earth's magnetic field is quite important as it protects the surface from charged particles and ions that are emitted in the Solar Wind. During active cycles of the Sun, solar flares erupt which can created intense geomagnetic storms in the upper reaches of the earth's ionosphere. The major effect for inhabitants of the Earth is usually a spectacular show of northern lights.

The magnetic field on the earth is a complex phenomena in which the spinning liquid outer core of the earth acts as a dynamo. For reasons yet to be fully understood, this dynamo changes its polarity periodically over geological time so that Magnetic North its sometimes aligned with geographic south and vice versa.

Stars too have magnetic fields. Sunspots, for instance, are large magnetic disturbances on the surface of the Sun. Pulsars are rapidly rotating neutron stars of radius 10 km. They have intense magnetic fields as a result of stellar collapse and a tremendous reduction in the radius. The field lines are so intense that they easily generate synchrotron radiation which can only escape at the the magnetic poles. If these poles sweep buy an observer, the observer sees pulses of radiation due to the rapid rotation (many times per second) of the neutron star.

Magnetic Force on a Moving Charge

One of the interesting aspects of magnetism is that you can't really measure its effects without having a moving charge. Let's envision a beam of electrons moving down a cathode ray tube. We apply a uniform magnetic field to that electron beam. IF we direct this electron beam into the field at different angles and velocities we invariably find that force on the elector beam is always perpendicular to the velocity of the individual electrons.

This is not the way other forces behave, such as gravity or the Coulomb force.

If we interchange the poles of the magnetic field, we will find that although the force changes direction, it remains perpendicular to the velocity. from our earlier work on vectors, the cross product of two vectors A and B is always perpendicular to both vectors:

The observed deflection of the beam, perpendicular to the velocity of the electrons, strongly suggests the following definition of the magnetic force and field:

You might also suspect that the total force is dependent on the total charge. That leads to the following definition for magnetic force:

This is more complicated math than is required for this level so we will simply approach this qualitatively. What is important here is that the total magnetic force depends on three things:

the total charge
the velocity of the charges
the strength of the magnetic field

If we return to the previous equation we can see what the units of the magnetic field B should be:

This funny combination of units is called a Tesla. A more common unit for magnetic field is called a Gauss which is 10^-4 Tesla. The Earth's magnetic field is about 0.5 Gauss. The magnetic field of a neutron star is approximately 10¹² Gauss. The magnetic field in your body is about 10^-8 Gauss.

Motion of a Charged Particle in a Magnetic Field

The key feature of the magnetic force exerted on a charged particle is that it is perpendicular to the velocity of that particle.

No matter what the orientation of the velocity is with respect to the field, the only force the particle feels in perpendicular to its direction of motion.

Consider the following diagram. We let a positively charged particle have a velocity along the x-axis, and apply a uniform magnetic field along the y-axis:

Using the right hand rule, which STAN will now demonstrate:

we can see that the magnetic force must be along the +z axis, as indicated in the figure. We can then ask what kind of motion the particle will undergo. Since the magnetic field is perpendicular to the velocity, the motion will be circular as the force is always oriented "inwards". This is a centripetal force as shown below:

This produces a trajectory which is a circle. As there will be no force along the magnetic field, the motion of the particle in that direction must be characterized by constant velocity as this component is unaffected by the magnetic field. An object that is therefore moving through a magnetic field, with a circular trajectory caused by a force perpendicular to its travel will undergo helical motion:

Visualization