Chapter 29

Chapter 29Magnetic FieldsIntroductionKnowledge of Magnetism and application dates back to 13th Century BCE in China.Magnetic Compass Needle (of Arabic/Indian invention)Greeks discovered magnetism ~800 BCE.Magnetite (Fe3O4)attracts iron. 1269, Pierre de Maricourt discovered that magnets have two poles, N and S, which can attract and repel one another. Introduction1600, William Gilbert suggested the Earth itself is a magnet. 1750, experimenters collectively show the attractive/repulsive forces follow the inverse square law.Very similar behavior to electric chargesA single magnetic pole (monopole) has not been isolated. Introduction1819, Hans Christian Oersted, discovered the link between electricity and magnetism. Current in wire caused deflection of a compass needle. 1820s, Michael Faraday and Joseph Henry (independently) showed that a changing magnetic field creates an electric field. Introduction1860s, James Clerk Maxwell, theoretically shows the reverse.A changing electric field causes a magnetic field.

This Chapter will focus on the effects of magnetic field on charges and current carrying wires. We will identify sources of magnetic field in Ch 30. 29.1 Magnetic Fields and ForcesRemember that an electric field surrounds any electric charge.

A magnetic field also surrounds any moving charge, and any permanent magnet.

29.1Magnetic field is a vector quantity.Represented by the symbol BThe vector direction of B aligns with the direction that the needle of a compass would point. Fields can be represented with magnetic field lines.29.1A typical bar magnetic field lines.A compass can be used in the presence of a magnet to trace the field lines.

29.1Iron filings are also useful (but messy) to identify magnetic field patterns.

29.1

29.1We can define a magnetic field B, at some point in space, in terms of the magnetic force FB, that the field exerts on a charged particle moving through the space with velocity v.

For now we will assume there are no electric/gravitational fields in the space. 29.1Experiments have shown thatThe magnitude of FB is proportional the charge q and speed v of the test particle.The magnitude and direction of FB depend on the velocity and the magnitude and direction of BWhen a charged particle moves parallel to the magnetic field vector, the value of FB is zero. 29.1Experiments show contd:At any angle 0, FB acts perpendicular to both v and B. The direction of magnetic force on a positive charge is opposite to the force on a negative charge.The magnitude of FB is proportional to sin29.1We can summarize these observations with the expression

29.1Remember the direction of the cross product v x B is determined by the Right Hand RulePoint your fingers in the direction of v.Curl them in the direction of B.Thumb points in the direction of FB.Remember the magnitude of a cross product

29.1

29.1From this we see that FB = 0 when = 0, 180oElectric field DifferencesFE acts along field lines, FB perpendicularFE acts on any particle w/ charge, FB only acts on moving charges.FE does work displacing a charged particle, FB does no work, (perpendicular force)29.1By rearranging the equation we determine units for magnetic field should be

The SI unit for magnetic field is the tesla (T) also given as

and

29.1Common B Fields

29.1Quick Quizzes p. 899Example 29.1 For Reference

29.2 Magnetic Force on a Current Carrying Conductor If a Force acts on a single charge moving through a B-Field, then it should also act on a wire with current placed in a B-Field. We can demonstrate this by looking at a wire suspended between the poles of a magnet. 29.2When there is no current in the wire, there is no force on the wire.

29.2When the current flows upward through the wire, the force causes a deflection to the left.

29.2When the current flows down through the wire, there is a deflection to right. 29.2To quantify these observations we will look at a wire segment of length L, cross-sectional area A, carrying a current I, in a uniform magnetic field B.

29.2We see that the force acting on a single moving charge (traditional current) follows the equation

If we multiple this force by the number of charges moving through the segment given as nAL, we have the force on the whole segment.

29.2Now remember that

So we can rewrite the expression as

Where L is a vector that points in the direction of the current and has a magnitude equal to the length of the wire segment.

29.2This expression only applies for a straight wire segment passing through a uniform B-Field.

29.2Now consider an uniformly shaped, but arbitrarily bent wire, in a B-Field.

29.2The force on any small segment will be

We integrate from end points a and b to find the total force on the wire

29.2The quantity represents the vector sum of each small segment, which will equal the vector length from point a to point b, L

29.2So we can conclude that the magnetic force on a curved current carrying wire, in a uniform B-field is equal to that on a straight wire connecting the end points and carrying the same current.

29.2We also note that if our conductor is a closed loop, we take the integral over the entire loop

But since the vector sum of a closed loop is zero

29.2Quick Quizzes p. 903Example 29.229.3 Torque on a Current Loop in a Uniform Magnetic FieldConsider the rectangular loop carrying current I in the figure below.Which sides have the magnetic force acting on them?Sides 2 and 4What directions do the forces act?

29.3We see that the magnitude of the forces are equal,Since the forces act on opposite sides inopposing direction, they have equivalent torque around the central axis.

29.3The maximum torque is given as

This can be simplified to

29.3If the loop is angled in the B-Field as shownThe lever arm for each torque is

So the overall magnitude of the torque is

29.3This can easily be expressed in vector notation as

The direction of vector A points perpendicular to the area of the loop following the RH Rule. Curl your fingers around the loop in the direction of the current. Thumb Points out A direction.

29.3Often the product IA is referred to as the magnetic dipole moment, or magnetic moment of the loop.

The magnetic moment has units of A.m2 and points in the same direction as A.

29.3For a coil of wire, with many turns

The potential energy associated with a loop of wire is given as

29.3From the expression, Umin = -B, when and B point in the same direction and Umax = +B, when and B point in opposite directions.

Quick Quizzes p 906Example 29.3, 29.429.4 The Motion of a Charged Particle in a Uniform Magnetic FieldRemember from 29.1, the magnetic force that acts on a particle is always perpendicular to the velocity (and therefore does no work).

If a positively charged particle moves with velocity v, perpendicular to a magnetic field B, what shape will its path take? 29.4Circular Path

29.4Using what we know about circular motion, and centripetal acceleration, we can find the radius of the circular path.

29.4We can define the angular speed (also called the cyclotron frequency)

The period of the cycle is given as

29.4If the velocity is not perpendicular to B, but some arbitrary angle, the perpendicular component causes circular motion, but the parallel component induces no force.The path is helical. 29.4In non-uniform fields the motion is complex. Particles can be come trapped oscillating back and forth.

29.4Trapping effect is often referred to as a magnetic bottle. This effect is shown in the Van Allen radiation belts, surrounding the earth, and is responsible for the Auroras Borealis and Australis .29.4Van Allen Belts

29.4Quick Quizzes p. 908Examples 29.6, 29.7

29.5 Applications of Charged Particles moving in Magnetic FieldsCharged particles moving in both Electric and Magnetic FieldsThe net force on the particle is generally sum of the Electric and Magnetic Forces

29.5Velocity Selector-Used to guarantee that only particles of a specific velocity enter an experiment.Consists of uniform and perpendicular electric and magnetic fields.

29.5As in the picture, the positively charged particle experiences an Electric force down, and magnetic force up. Only when the velocity is correct will the forces balance (net force zero) and the particle will pass straight through. Particles with incorrect speeds will deflect either up or down. 29.5Setting equal to zero

V is given as the ratio of E to B

29.5Mass Spectrometer- a method for isolating ions of different masses/charges. Particles begin by passing through a velocity selector, then pass though an additional magnetic field.

29.5The greater the mass of the particle, the wider the radius of arc it will travel, creating a spectrum of masses on the detection screen.Positively charged particles will deflect to one side, negatively charged particles will deflect to the other. 29.5We can determine the ratio of m/q of unknown particles by rearranging the radius equation

And we know the velocity from the selector so

29.5Using this information we can determine the mass ratios of various isotopes of a given ion. Can be used for collecting fairly pure samples of specific isotopes it does not scale well to industrial levels.

Quick Quiz p. 912

29.5A variation of the concept was completed by J.J. Thomson in 1897 to verify the ratio of e/me for electrons, helping to confirm their existence.

29.5Cyclotron- a device used to accelerate charged particle to high speeds often to bombard atomic nuclei to produce various nuclear reactions. Depends on both electric and magnetic fields.

29.5The source P, emits charged particles. The magnetic field causes the particles to travel a circular path, with a period of T. The circular path is divided into two semicircular sections dees, with an alternating potential difference across the gap. 29.5Every time the charged particle crosses the gap it accelerates, and the potential difference flips by the time it comes halfway around. Each gap acceleration widens the circular path due to the increased velocity. The K gained across the gap is equal to qV.29.5The kinetic energy of the particles upon exiting the cyclotron is given as

And is often measured in eV (electron volts), keV and MeV. (1 eV = 1.6 x 10-19 J)With an upper limit of 20 MeV before the effects of relativity come into play.

29.6 The Hall EffectWhen a current carrying conductor is placed in a magnetic field, a potential difference across the conductor is created. This comes from the deflection of charge carriers due to the magnetic force.The potential difference is known as the Hall Voltage, and helps determine the sign of the charge carriers, and is also a method for measuring magnetic fields.

29.6An easy way to demonstrate the Hall Effect is to look at a flat rectangular shaped conductor.The perpendicular magnetic field creates a magnetic force that causes the charge carriers to deflect to the top of the conductor.29.6The deflection of carriers to the top, leaves a deficit of carriers on the bottom side, resulting in an electric field (Hall Field) and Voltage across the conductor.

29.6Eventually equilibrium is reached and there is no further deflection. The upward magnetic force is balanced by the resulting downward Electric Force.

A sensitive voltmeter can measure the potential difference and its polarity gives the sign of the charge carrier. 29.6

29.6At equilibrium

So the Hall Field is

And the Hall Voltage is

29.6Since the drift velocity is directly related to charge density and current

So the Hall Voltage can be given as

29.6One last adjustment, A/d represents the thickness of the conductor, t

Everything in this expression can be measured directly, except for 1/nq which is named the Hall Coefficient, RH.

29.6Since we can measure everything else, determining the Hall Coefficient easily gives the sign and density of charge carriers.It allows us to confirm that many conducting metals give up one electron per atom.Certain metals and semiconductors give up much fewer. 29.6Medical Application- because of the ions carried by in the blood stream, the Hall effect is used by electromagnetic blood flowmeters.The diameter of the artery is measured, a magnetic field is applied, then the Hall voltage is measured. From this vd can be determined.

Example 29.8 p. 916