Magnetism

Magnetic FieldYou are all familiar with the behavior of a magnet: a piece of iron placed near a magnet will feel a force of attraction to the magnet.The magnetic force is not a contact force; the piece of iron and the magnet need not be touching? How does the iron know that the magnet is present?A magnet creates a magnetic field in the space surrounding it. The iron experiences the magnetic field set up by the magnet.All magnets have a “north pole” and a south pole”. Magnetic field lines leave north poles and enter south poles. The field set up by a bar magnetic looks like:

Field lines begin on north poles and end on south poles.The field is strongest where the field lines are the most dense.

Magnetic field is tangential to field linesB

Units of magnetic field: Tesla

All the planets with molten cores have magnetic fields. The magnetic field of Earth looks like

The direction of the Earth field has reversed itself many times in the past.Currently, field lines leave the region of Antarctica and return in the Arctic. This makes the geographic North Pole a magnetic south pole.

Magnetic Force

You have learned that a charge in an electric field experiences a force: EqF

Does the electric force depend on which way the charge is moving?Does the electric force depend on how fast the charge is moving?The behavior of the magnetic force is much different.

1. A charge must be moving to feel a magnetic force. Even if the field is very strong, a charge that is at rest will feel no force.

2. A charge must be moving in such a way that it cuts across the field lines to feel a magnetic force. If the charge moves parallel to the field lines, it feels no force.

3. When a charge cuts across field lines, the faster it moves the bigger the force it will feel.

4. For a given speed, the maximum force a charge will feel occurs when it cuts directly across the field lines, that is, it moves perpendicular to the field lines.

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This charge feels no force.

This charge fees the maximum force.

When a charge moves perpendicular to the field, the magnitude (size) of the force it feels is given by the equation:

Exercise: An electron moving in a magnetic field of 0.35 T experiences a force of magnitude 5.2 x 10-15 N. Calculate the speed of the electron.

You might well ask in which direction is the force. Let’s approach this in two steps. 1. The magnetic force on a moving charge is always perpendicular to both the

velocity and the magnetic field.

Two vectors that are not parallel always define a geometric plane. This implies that the magnetic force is always perpendicular to the plane defined by the velocity vector and the magnetic field vector.However, there are two ways a vector can be perpendicular to a plane. Which one is it?The answer is given by the so called right hand rule:

2. To determine the direction of the magnetic force on a moving + charge, place the fingers of the right hand in the direction of the magnetic field. Rotate your hand so that the thumb of your right hand is in the direction of the velocity. Then your palm will point in the direction of the force.

To determine the force on a – charge, use the right hand rule and then reverse the answer.

Because of the nature of the magnetic force, you will have to start to think more in three dimensions. When we draw vectors let’s use the following conventions:

When a vector lies in the plane of the page, use an arrow to represent the vector (this is what we have done until now.

When a vector points into the paper, use an “X” to represent the vector.When a vector points out of the paper, use an “O”

Exercises: Determine the direction of the magnetic force in the examples below.

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The magnetic force is always perpendicular to the velocity. This means that there is never a component of the force either in the same direction as the velocity, or opposite to the velocity.Therefore, the magnetic force can never cause a charged particle to either speed up or slow down.The perpendicular force can cause the velocity to change direction, but this is the only way the magnetic force can produce accelerations on moving charges. Another way to say this is that the magnetic force always does zero work on a moving charge.

So, what exactly happens when charges move in a magnetic field?Let’s look at the possible motion paths when charges enter a uniform magnetic field. There will be three basic types of paths (trajectories).

Case 1: Velocity and magnetic field are parallel or anti parallel. Bv

║In this case the force is zero. Inertia then tells you that the particle will either remain at rest if it is currently at rest, or remain moving at a constant velocity.

Case 2: Velocity is perpendicular to the magnetic field. Bv

In this case the charge will feel the maximum magnetic force

The following figure illustrates what happens.

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The charge will move in a circle.The magnetic force supplies the necessary centripetal force. Applying Newton’s 2nd law, you have

Example: Calculate the radius of the circular orbit of an electron moving at 3 x 106 m/s perpendicular to a magnetic field of 0.5 T

What happens ifa. field doubles?b. speed triples?

Case 3. Velocity is neither parallel nor perpendicular to the field. This means that there will be a component of the velocity parallel to the field v║ and a component of the velocity that is perpendicular to the field v┴ . If only v║ was present, the charge would just move at a constant speed in a straight line. If only v┴ was present, the charge would move in a circle around the field lines. Since both are present, you get both motions at once: a helix with the axis of the helix along the field lines.

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Earth is constantly bombarded by high energy protons streaming off the Sun (solar wind).The magnetic field of the Earth deflects these particles, diverting them to the pole regions as they spiral around the field lines. When they hit the upper atmosphere, the collisions release light energy producing the aurora.

Force on a Current

You have seen that moving charges experience a force in a magnetic field. All of the examples given so far are rather exotic and not readily available to see in everyday life.However, an electric current moving though a wire is a readily available source of moving charges and can easily be seen to feel a force in a magnetic field.The process can be analyzed as shown below:

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The total charge Nq will move past the right cross section of wire in a time

Using the definition of current , you can write:

Notice that in the last equation, the product Nqv appears, and this also appears in the equation for the total force on the wire segment. Substituting, you get

Force on a current segment in a magnetic field.

Exercise: The 50 cm segment of wire shown below experiences a force of 7 N directed up when a current of 4 A is established in the segment. Determine the magnitude and direction of the magnetic field present.

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Using the right hand rule, you can see that the field must point into the paper.Use the formula to calculate the field strength.

The magnetic force acting on a current carrying wire is the basis for the operation of an electric motor. A common configuration has a coil of wire pivoted between the poles of a magnetic as shown below.

Using the right hand rule, you can verify that the opposite sides of the coil feel forces that will cause the coil to rotate.

Origin of the Magnetic Field

Up until now, you have been assuming that magnetic fields affect charges and you have used the force law to investigate the behavior of the moving charges. But where does the magnetic field come from in the first place?Well, you could say it comes from magnets, but magnets are not actually the fundamental source of magnetism, and it is not at all clear how a magnet creates a field anyway.Just as electric fields are created by charges, so to are magnetic fields. However, to create a magnetic field, a charge must be moving.Note the symmetry in this: only moving charges can feel a magnetic force and only moving charges can create a magnetic field. The simplest configuration of moving charges that we can study is a long wire wire carrying a constant current.

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Field lines for a long wire carrying current into page.

Such a current creates field lines that form closed circles centered on the wire.From the side, the field goes into the page on one side of the wire, and out of the page on the other.

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You can determine the direction of the field due to a long current carrying wire using another right hand rule:

Place the thumb of the right hand in the direction of the current. The fingers will curl in the direction of the field lines.

The strength of the magnetic field depends on the distance from the wire and the size of the current. The formula is:

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Exercise: Calculate the field 20 cm from a long wire carrying a current of 5 A.

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What is the direction of the field?

FerromagnetismIf moving charges are the source of magnetic fields, what are the moving charges in everyday

iron magnets?1) Orbital motion of electrons

While this motion does create magnetic fields, over a scale much larger than an individual atom, it will average out to zero since different atoms will have their electrons circulating in different directions.

2) Spin: electrons have an intrinsic spin; this motion will create magnetic fields also. Once again, over a large scale, these fields tend to cancel, except for certain materials.In some materials there is a residual interaction between the spinning electrons that tends to make them spin in same direction, and this causes the magnetic fields they create to add up, not cancel. These materials are said to be “ferromagnetic” and include iron, cobalt, nickel, and others.

The interaction between spinning electrons has a finite range, and one finds that the tendency to line up is limited to a region large compared to atoms but still small compared to our everyday scale.These regions are called “domains”.To actually make a magnet, one has to align the domains, perhaps with a large external field.

iron wisker

ferromagnetic garnet film

A species of marine bacteria that live in the sediment at the bottom of the ocean actually contain roughly 5-20 magnetic particles aligned along the long axis of the organism. Should they be disturbed out of the sediment, their internal magnet aligns along the local earth field line and they swim back down to the sediment.In the northern hemisphere, the organism has a south pole at its anterior, while the opposite occurs in the southern hemisphere.A northern hemisphere organism brought to the southern hemisphere will swim to the surface!

Electromagnetic Induction

You know that a conductor is a material that contains many mobile electrons.Suppose you move a conducting rod through a magnetic field?Each mobile electron will feel a force tending to push them all toward one end of the rod.

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However, once electrons start to pile up on one end, leaving excess positive charge on the other end, it gets harder and harder to move more electrons down.The two ends of the rod become polarized and these separated charges create an electric field in the rod.Equilibrium is soon established. The magnetic force down will just equal the electric force up on each electron.

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Electric field created by moving conductor in a magnetic field.

The electric field created in a rod in this manner is uniform. This means we can use the formula for the voltage difference between two points in a uniform field.

One says that the motion of the conductor in the magnetic field has created an induced voltage.In a sense, while the conductor is moving, the two ends of a moving conducting rod become like the terminals of a battery.

Example: Calculate the magnetic field strength required in induce a voltage difference of 1.5 V between the ends of a conducting rod of length 0.5 m moving at a speed of 10 m/s through a magnetic field. Indicate the + and – ends of the conductor.

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If the moving conductor is allowed to slide on conducting rails, the voltage difference created by the movement can produce an electric current, an induced current. This is the basic idea of an electric generator : mechanical energy is put into the system to move the conductor, and this mechanical energy is then converted to electrical energy.

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You can predict the direction of the current by asking yourself which way the force will act on a hypothetical + charge in the moving rail.

As long as the rail continues to move, a current will flow. A very interesting thing occurs once the current starts to flow: the magnetic field tries to stop it!Use the right hand rule to determine the direction of the force on the current in the rail.The motion of the conductor in the field is what created the current, but as soon as the current starts to flow, the magnetic force on the current opposes the motion that caused the current to flow in the 1st place.This always happens to an induced current.

Lenz’s Law: An induced current will always flow in such a way as to oppose the change that caused it.

Exercise: Use the right hand rule (twice) to verify that Lenz’s Law is true for the generator shown below. Hint: 1st find the direction of the induced current, then find the force on that current.

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The rail system studied on the previous two slides is perhaps the simplest form of generator.Michael Faraday found that the induction process occurs whenever the number of magnetic field lines cutting through a loop changes.For example, if you rotate a loop in a magnetic field, the figure below shows that the number of field lines cutting through the loop changes. This means an induced current will flow while the change is occurring.

Field lines cut through loopField lines do not cut through loop

Question: Explain why a current flows in the rail system in terms of Faraday’s Law.

Both the rail system and the rotating loop involve changing the area of the loop through which the field cuts. However, you can also induce a current by keeping the area the same but changing the magnetic field strength. For example, if you move a magnet toward a loop, you increase the number of field lines cutting though the loop, and this will cause a current to flow.

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This is the most general statement of Faraday’s Law: A changing magnetic field creates an electric field.Symbolically you can write:

The electrons in the loop experience a magnetic field that changes in time.This produces an electric field that then causes the current to flow. Even if the loop was not present, the electric field would still be created by the changing magnetic field.

Faraday’s Law

Electromagnetic Radiation

Faraday’s Law gives the magnetic field a special place: by changing the magnetic field, you can create an electric field. Physics abounds with the concept of symmetry and it was not too long before the inquisitive James Clerk Maxwell wondered if the magnetic field was really so very “special”.Maxwell speculated that changing electric fields could in fact create magnetic fields. Symbolically you could write:

Maxwell predicted that if either an electric field or a magnetic field changed, the resulting interplay between the changing fields would produce a disturbance that would leapfrog away from the original source of the disturbance. This disturbance is called electromagnetic radiation.Electromagnetic radiation consists of changing electric and magnetic fields that create each other through the interaction of Faraday’s and Maxwell’s laws.

Maxwell’s Law

The number of times per second that the fields change is called the frequency of the EM radiation. Accounting for all the ranges in frequency, you have the electromagnetic spectrum:

Increasing frequency

E-M Radiation created by a changing current

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