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MAGNETISM

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MAGNETISM

Learning goals

We will first study the second stage in the magnetic interaction—that is, how movingcharges and currents respond to magnetic fields. In particular, we will see how tocalculate magnetic forces and torques, and we will discover why magnets can pick upiron objects like paper clips.

Then, we will complete our picture of the magnetic interaction by examining how moving charges and currents produce magnetic fields.

We saw in Electricity that the electric force arises in two stages:

(1) a charge produces an electric field in the space around it.

(2) a second charge responds to this field.

Magnetic forces also arise in two stages:

(1) a moving charge or a collection of moving charges (that is, an electric current) produces a magnetic field.

(2) A second current or moving charge responds to this magnetic field,and so experiences a magnetic force.

1. Magnetism

Magnetic phenomena were first observed at least 2500 years ago in fragments of magnetized iron ore found near the ancient city of Magnesia (Turkey).

Since ancient times, certain materials, called magnets, have been known to have the property of attracting tiny pieces of metal. This attractive property is called magnetism.

Magnetite:Fe3O4

The magnetic compass is an old Chinese invention, probably first made in China during the Qin dynasty (221-206 B.C.) Chinese fortune tellers used lodestones (a mineral composed of an iron oxide which aligns itself in a north-south direction) to construct their fortune telling boards. Eventually someone noticed that the lodestones were better at pointing out real directions, leading to the first compasses.

Compass used in Navigation (the fore father of GPS was the advent of the compass and sextant.

Needle compass for magnetism experiments

Today, magnets/magnetism are everywhere

First description of magnets and their interaction: in terms of Magnetic Poles

The strength of a magnet is concentrated at the ends, called North and South “poles” of the magnet.

North pole of a magnet pointed toward a compass.

South pole of a magnet pointed toward a compass.

Magnetic interactions between poles

Either pole of a bar magnet attracts an unmagnetized object that containsiron, such as a nail.

The earth itself is a magnet. Its north geographic pole is close to a magnetic south pole, which is why the north pole of a compass needle points north

The field, which is caused by currents in the earth’s molten core, changes with time; geologic evidence shows that it reverses direction entirely at irregular intervals of 104 to 106 years.

Magnetic Poles Versus Electric Charge

The concept of magnetic poles may appear similar to that of electric charge, andnorth and south poles may seem analogous to positive and negative charge.

Magnetic poles cannot be separated

Magnetic monopoles do not exist in isolation

dipole magnet

NS

However:

The first evidence of the relationship of magnetism to moving charges was discovered in 1820 by the Danish scientist Hans Christian Oersted.

a compass needle was deflected by a current-carrying wire

Later, Michael Faraday in England and Joseph Henry in the United States discoveredthat moving a magnet near a conducting loop can cause a current in the loop.

Electric and magnetic interactions prove to be intimately connected. – see Maxwell’s equations

2. Magnetic Field

We can describe magnetic interactions in a similar way:

Before: formulation of electric interactions => the concept of electric field.

2-step mechanism

Quick Note on Magnetic Fields

Like the electric field, the magnetic field is a Vector, having both direction and magnitude

We denote the magnetic field with the symbol

The unit for the magnetic field is the tesla

mAN /111 Ttesla

There is another unit that is also used and that is the gauss

Tgauss 4101

Unlike Electric Fields which begin and end on charges,Magnetic Fields have neither a beginning nor an end

ExamplesTerrestrial B field ~ 4x10-5 TSolenoid ~ 10-3 TPermanent magnet ~ 10-1 TAtomic interactions ~ 10 TSuperconducting magnet ~ 102 TWhite dwarfs ~ 102 - 103 TNeutron stars < 108 T

Magnetic Field Lines

We can describe magnetic field lines by imagining a tiny compass placed at nearby points.

The direction of the magnetic field B at any point is the same as the direction indicated by this compass.

Field B is strong where lines are dense and weak where lines are sparse.

Field lines:go out from Nenter in Scontinue inside of the magnet

3. Magnetic Field Lines and Magnetic Flux

Magnetic field lines produced by some common sources of magnetic field

(a) Like little compass needles, iron filings line up tangent to magnetic field lines. (b) Drawing of the field lines for the situation shown in (a).

Visualization

Magnetic Flux and Gauss’s Law for Magnetism

We define the magnetic flux B through a surface just as we defined electric fluxin connection with Gauss’s law:

Flux unit is called the weber (1 Wb)

B varies with the position:

Across unit area dA B constant:

In Gauss’s law the total electric flux through a closed surface is proportionalto the total electric charge enclosed by the surface.

By analogy, if there were such a thing as a single magnetic charge (magnetic monopole), the total magnetic flux through a closed surface would be proportional to the total magnetic charge enclosed. But we have mentioned that no magnetic monopole has ever been ever observed.

The total magnetic flux through a closed surface is always zero.

Gauss’s law for magnetism

States out the absence of magnetic monopoles

Second eq. from the set of 4 Eqs. of Maxwell

Electric dipole Magnetic dipole (or current loop)

Gauss’ Law for Magnetism

4. Magnetic Forces on Moving Charges

Lorentz force

Finding the direction of the magnetic force on a moving charged particle.

Key Concepts Force is perpendicular to field direction and velocity Therefore, magnetic fields do no work on particles There is no change in magnitude of velocity, just direction

Motion of a Point Charge in a Magnetic Field

Cyclotron period

Cyclotron frequency

qB

mvr

qB

mT

2

m

qB

Tf

21

Motion due to a Magnetic Force

The combination of these two motions results in a helical type motion

What is the motion like if the velocity is not perpendicular to B?

We break the velocity into components along the magnetic field and perpendicular to the magnetic field

The component of the velocity perpendicular to the magnetic field will still produce circular motion

The component of the velocity parallel to the field produces no force and this motion is unaffected

qB

mvr

The Van Allen radiation belts around the earth. Near the poles, charged particles from these belts can enter the atmosphere, producing the aurora borealis (“northern lights”) and aurora australis (“southern lights”). (b) A photograph of the aurora borealis.

When a charged particle moves through a region of space where both electricand magnetic fields are present, both fields exert forces on the particle.

Velocity selector (Wien filter):

B

EvqEqvB

If the velocity of the charged particle is just right v=E/B then the net force on the charged particle will be zero => particle undeviatedSelects particles with given v from distribution

We know that a single moving charge experiences a force when it moves in a magnetic field

What is the net effect if we have multiple charges moving together, as a current in a wire?

Magnetic Force on a Current-Carrying Wire

Experiment

We start with a wire of length l and cross section area A in a magnetic field of strength B with the charges having a drift velocity of vd

The total number of charges in this section is then where n is the charge density

nAlThe force on a single charge moving with drift velocity vd is:

BqvF d

So the total force on this segment is

BlAvqnF d

We have so far that BlAvqnF d

But we also have that dvqnJ and AJI

Combining these, we then have that BlIF

The force on the wire is related to the current in the wire and the length of the wire in the magnetic field

If the field and the wire are not perpendicular:

The direction of l is the direction of the current

F IL B

For a wire of arbitrary shape

The magnetic force acting on the segment is

Thus, the total force is:

the magnetic force can be obtained by summing over the forces acting on the small segments that make up the wire.

A common application of the magnetic forces on a current-carrying wire is foundin loudspeakers

force on the voice coil that is proportional to the current in the coil; the direction of the force is either to the left or to the right, depending on the direction of the current.

Closed loop carrying a current I in a uniform magnetic field

Torque on a Current Loop

But: Torque non zero

Torques on Current Loops

• Key concept – a current loop experiences no net force in a uniform B field but does experience a torque

IaBFF 21

sin2bF sinIaBb sinIAB sinNIAB

Bμτ Magnetic dipole moment

nμ NIA

Potential Energy of a Current Loop

As the loop rotates because of the torque, the magnetic field does work on the loop

We can talk about the potential energy of the loop and this potential energy is given by

The potential energy is the least when µ and B are parallel and largest when µ and Bare antiparallel

U B

The Direct-Current Motor

In a dc motor a magnetic field exerts a torque on a current in the rotor.

The Hall Effect

There is another effect that occurs when a wire carrying a current is immersed in a magnetic field

Assume that it is the positive charges that are in motion

These positive charges will experience a force that will cause them to also move in the direction of the force towards the edge of the conductor, leaving an apparent negative charge at the opposite edge

The fact that the there is an apparent charge separation produces an electric field across the conductor

Eventually the electric field will be strong enough so that subsequent charges feel an equivalent force in the opposite direction

or

Since there is an electric field, there is a potential difference across the conductor which is given by

dBvdEV de

The Hall Effect

Hall voltageHall field sensorbut:

The Hall Effect allows us to determine the sign of the charges that actually make up the current

If the positive charges in fact constitute the current, then potential will be higher at the upper edge

If the negative charges in fact constitute the current, then potential will be higher at the lower edge

For a metal: experiment shows that the second case is true

The charge carriers are in fact the negative electrons

The Hall Effect dBvdEV de

Determination of the concentration of current-carrying charges in the material

The Hall effect can also be used for a direct measurement of electron driftspeed in metals.

dBvdEV de

From:

The Hall Effect

Hall field sensor

H H dV E d v Bd

II. SOURCES OF MAGNETIC FIELD

We studied the forces exerted on moving charges and on current carryingconductors in a magnetic field.

But how are magnetic fields created? We know that both permanent magnets and electric currents in electromagnets create magnetic fields.

We’ve learned that a charge creates an electric field and that an electric fieldexerts a force on a charge.

But a magnetic field exerts a force only on a moving charge.

Also a charge creates a magnetic field only when the charge is moving.

1st step: the magnetic field of a single point charge q moving with a constant velocity

2nd step: the magnetic field of a collection of charges nq <=> current density j=nq

1. Magnetic Field of a Moving Charge

Exp

μo= permeability of free space μo= 4 x 10-7 T·m·A-1

2. Magnetic Field of a Current Element

Principle of superposition of magnetic fields:The total magnetic field caused by several moving charges is the vector sum ofthe fields caused by the individual charges.

I

laws of Biot and Savart

total magnetic field at any point in space due to the current in a complete circuit. integrate over all segments

𝑑𝑙 that carry current; symbolically

Ex. Magnetic Field of a Straight Current-Carrying Conductor

From Biot and Savart

When x<<a (infinite long conductor) =>

at all points on a circle of radius r around the conductor, the magnitude B is:

3. Force Between Parallel Conductors

Segments of two long, straight, parallel conductors separated by a distance r and carrying currents I and I’ in the same direction.

Each conductor lies in the magnetic fieldset up by the other, so each experiences a force.

Two parallel conductors carrying current in the same direction attract each other. Parallel conductors carrying currents in opposite directions repel each other.

Magnetic Forces and Defining the Ampere

The attraction or repulsion between two straight, parallel, current-carrying conductorsis the basis of the official SI definition of the ampere:

4. Ampere’s Law

like Gauss’ law for electric field, uses symmetry to calculate B field around a closed curve C

I

long, straight, current-carrying conductor

Field of a long cylindrical conductor

A circular integration path outside the conductor encloses the total current I in the conductor, so:

Field of a solenoid

In a length L there are nL turns, each of which passes once through abcd carrying current I.