Chapter 29

Magnetic Fields

A Partial History of Magnetism

13th century BC

Chinese used a compass

Uses a magnetic needle

Probably an invention of Arabic or Indian origin

800 BC

Greeks

Discovered magnetite (Fe3O4) attracts iron

A Partial History of Magnetism

Hans Christian Oersted

Danish Physicist

Discovered the relationship between electricity and magnetism in 1819

during a lecture demonstration!

An electric current in a wire deflected a nearby compass needle

Demo Ec4:

Oersted’s Experiment

When a current flows through the conductor the compass needle deflects due to the magnetic field set up by the current.

Reversing the direction of the current reverses the direction of the magnetic field.

Oersted in 1819 revolutionised the study of electricity & magnetism, sparking scientists to investigate the subject. He himself couldn’t set up experiments and had to rely on assistants. He got left behind!

Demo Ec1: Lines of force

in a magnetic field

Iron filings sprinkled

onto perspex sheet

on top of bar

magnet

Magnetic Poles

Every magnet, regardless of its shape, has two poles

Called north and south poles

Poles exert forces on one another

Similar to the way electric charges exert forces on each other

Like poles repel

N-N or S-S Unlike poles attract

N-S

Magnetic Poles, cont.

The poles received their names due to the way

a magnet behaves in the Earth’s magnetic field

If a bar magnet is suspended so that it can

move freely, it will rotate

Magnetic north pole points toward the Earth’s north

geographic pole

i.e. Earth’s north geographic pole is a magnetic

south pole

Similarly, the Earth’s south geographic pole is a

magnetic north pole

Magnetic Poles, final

The force between two poles varies as

the inverse square of the distance

between them

A single magnetic pole has never been

isolated

i.e. magnetic poles always found in pairs

There is some theoretical basis for the

existence of monopoles – single poles

Magnetic Fields

(magnetic induction)

A vector quantity

Symbol B

Direction given by that which the north

pole of a compass needle points

Magnetic field lines trace this direction

in space

MFA02VD1: Compass needle

shows direction of magnetic

field around a bar magnet

Magnetic Field Lines

for a Bar Magnet

Compass can be

used to trace the

field lines

The lines outside

the magnet point

from the North pole

to the South pole

MFM02VD1: Magnetic Field

lines around bar magnets

Magnetic Field Lines,

Bar Magnet

Iron filings are used

to show the pattern

of the magnetic field

lines

The direction of the

field is the direction

a north pole would

point

Magnetic Field Lines,

Unlike Poles

Iron filings are used to

show the pattern of

the electric field lines

The direction of the

field is the direction a

north pole would point

c.f. electric field

produced by an

electric dipole

Magnetic Field Lines,

Like Poles

Iron filings are used to show the pattern of the electric field lines

The direction of the field is the direction a north pole would point

c.f. electric field produced by like charges

Magnetic Fields and Forces

The magnetic field, B, at some point in

space can be defined in terms of the

magnetic force, FB

Note: B is sometimes known as the

Magnetic Induction

A magnetic force will be exerted on a

charged particle moving in a magnetic

field

FB on a Charge Moving

in a Magnetic Field

Magnitude proportional to charge and speed of the particle

Direction depends on the velocity of the particle and the direction of the magnetic field It is perpendicular to both

FB = q v x B FB is the magnetic force

q is the charge

v is the velocity of the moving charge

B is the magnetic field

Direction

FB perpendicular to plane formed by v & B

Oppositely directed forces are exerted on

charges of different signs

cause the particles to move in opposite directions

Direction given by

Right-Hand Rule

Fingers point in the

direction of v

(for positive charge;

opposite direction if

negative)

Curl fingers in the

direction of B

Then thumb points in the

direction of v x B; i.e. the

direction of FB

The Magnitude of F

The magnitude of the magnetic force on

a charged particle is FB = |q| vB sin

is the angle between v and B

FB is zero when v and B are parallel or

antiparallel

= 0° or 180°

FB is a maximum when v and B are

perpendicular

= 90°

Example

Proton moves at v = 8x106 m/s in x-direction.

Enters region where B = 2.5 T directed at 60°

to x-axis in xy-plane. What is the initial force

on the proton?

Right Hand rule puts F in the z-direction.

F qvB sin( )

(1 .6 10 19

C)(8 106

m/s)(2.5 T)sin(60 )

2.77 10 12

N . Tiny!

Differences Between Electric

and Magnetic Fields

Direction of force

Electric force acts along direction of electric field

Magnetic force acts perpendicular to magnetic

field

Motion

Electric force acts on a charged particle

regardless of whether the particle is moving

Magnetic force acts on a charged particle only

when the particle is in motion

Differences Between Electric

and Magnetic Fields

Work

Electric force does work in displacing a

charged particle

Magnetic force associated with a steady

magnetic field does no work when a

particle is displaced

Force is perpendicular to the

displacement, so F.ds=0

Work in Magnetic Fields

The kinetic energy of a charged particle

moving through a magnetic field cannot

be altered by the magnetic field alone

The field can only alter the direction of

the particle, not its speed

Units of Magnetic Field

The SI unit of magnetic field is the tesla (T)

Since FB = |q| vB sin

The cgs unit is a gauss (G)

1 T = 104 G

1 T 1N

C m/s 1

N

A m

Typical Magnetic Field Values

Quick Quiz 29.1

The north-pole end of a bar magnet is held near a positively

charged piece of plastic. The plastic is

(a) attracted

(b) repelled

(c) unaffected by the magnet

Answer: (c). The magnetic force exerted by a magnetic field

on a charge is proportional to the charge’s velocity relative

to the field. If the charge is stationary, as in this situation,

there is no magnetic force.

Quick Quiz 29.1

Quick Quiz 29.2

A charged particle moves with velocity v in a magnetic field

B. The magnetic force on the particle is a maximum when v

is:

(a) parallel to B

(b) perpendicular to B

(c) zero

Answer: (b). The maximum value of sin θ occurs for θ =

90°.

Quick Quiz 29.2

Quick Quiz 29.3

An electron moves in the plane of this paper toward the top

of the page. A magnetic field is also in the plane of the page

and directed toward the right. The direction of the magnetic

force on the electron is

(a) toward the top of the page

(b) toward the bottom of the page

(c) toward the left edge of the page

(d) toward the right edge of the page

(e) out of the page

(f) into the page

Answer: (e). Out of the page. The right-hand rule gives the

direction. Be sure to account for the negative charge on the

electron.

Quick Quiz 29.3

Magnetic Force on a Current

Carrying Conductor

A force is exerted on a current-carrying

wire placed in a magnetic field

The current is a collection of many charged

particles in motion

The direction of the force is given by the

right-hand rule

Ec9: Magnetic force on

current carrying conductor

Flexible wire placed between poles of a strong magnet. When current is passed through it, the wire dramatically jumps out.

Can demonstrate F is perpendicular to L and B by suspending wire in different orientations.

See also corridor display.

Note on Notation

The dots indicate the direction is out of the page

The dots represent the tips of the arrows coming toward you

The crosses indicate the direction is into the page

The crosses represent the feathered tails of the arrows

MFA03AN2: Force on a

current-carrying conductor

Force on a Wire

When no current

flows there is no

force

Therefore, the wire

remains vertical

Force on a Wire (2)

With B into the page, and

Current up the page, then

Force is to the left

Apply the right hand rule

Force on a Wire, equation

Magnetic force is exerted on each moving charge in the wire

F = q vd x B

Total force is the product of force on one charge times the number of charges

F = (q vd x B)nAL n is the number of charges

per unit volume

Force on a Wire, (4)

In terms of the current, this becomes F = I L x B

L is a vector that points in the direction of the current (i.e. of vD) Magnitude is the length L of the segment

I is the current = nqAvD

(think about the units to see this)

B is the magnetic field

Force on a Wire of

Arbitrary Shape

Consider a small

segment of the wire,

ds

The force exerted

on this segment is

F = I ds x B

The total force is

Ib

d a

F s B

Force on a Wire, Case 1

Suppose that B is uniform.

Then

becomes F=I L’xB

L’ is the vector sum of all the length elements from a to b

Thus, the magnetic force on a curved current-carrying wire in a uniform field is equal to that on a straight wire connecting the end points and carrying the same current

Ib

d a

F s B

Force on a Wire, Case 2

An arbitrary shaped closed

loop carrying current I in a

uniform magnetic field

The length elements form a

closed loop, so the vector sum is

zero

Thus, the net magnetic force

acting on any closed current

loop in a uniform magnetic

field is zero

FB I d s B 0 since d s 0

Quick Quiz 29.4

The four wires shown below all carry the same current from point A to

point B through the same magnetic field. In all four parts of the figure, the

points A and B are 10 cm apart. Which of the following choices ranks the

wires according to the magnitude

of the magnetic force exerted on

them, from greatest to least?

(a) a, b, c

(b) a, c, b

(c) a, c, d

(d) a, d, c

(d) b, c, d

(e) c, b, d

Answer: (c). The order is (a), (b) = (c), (d). The magnitude

of the force depends on the value of sin θ. The maximum

force occurs when the wire is perpendicular to the field (a),

and there is zero force when the wire is parallel (d). Choices

(b) and (c) represent the same force because a straight wire

between A and B will have the same force on it as the curved

wire for a uniform magnetic field.

Quick Quiz 29.4

Quick Quiz 29.5

A wire carries current in the plane of this paper towards the

top of the page. The wire experiences a magnetic force

toward the right edge of the page. The direction of the

magnetic field causing this force is

(a) in the plane of the page and toward the left edge

(b) in the plane of the page and toward the bottom edge

(c) out of the page

(d) into the page

Answer: (c). Use the right-hand rule to determine the

direction of the magnetic field.

Quick Quiz 29.5

Charged Particle in a

Magnetic Field

Consider a particle moving

in an external magnetic

field with its velocity

perpendicular to the field

The force is always

directed toward the centre

of the circular path

The magnetic force causes

a centripetal acceleration,

changing the direction of

the velocity of the particle

Force on a Charged Particle

Equating the magnetic and centripetal

forces:

Solving gives r = mv/qB

r is proportional to the momentum of the

particle and inversely proportional to the

charge and to the magnetic field

F qvB mv

2

r

More About Motion of

Charged Particle

The angular speed of the particle is

The angular speed, is also referred to as

the cyclotron frequency

The period of the motion is

v

rqB

m

T 2 r

v

2

2 m

qB

Van Allen Radiation Belts

The Van Allen radiation

belts consist of charged

particles surrounding the

Earth in doughnut-shaped

regions

“Cosmic ray” particles from

the Sun are trapped by the

Earth’s magnetic field

The particles spiral from

pole to pole

Can result in Auroras

The Earth’s Magnetosphere

Charged particles trapped in the Van Allen belts and create aurorae

Aurorae: MFM11VD1

Typically 100–400 km above the Earth

Ec5: Deflection of electron beam

by magnetic & electric fields

Use either a cathode

ray tube or electron

beam deflection tube.

Possible to determine

e/m by size of the

deflection.

Mark North pole of

magnet to show that

F = q v x B.

MFM03AN1: Lorentz force in

electric & magnetic fields

Charged Particles Moving in

Electric and Magnetic Fields

In many applications, charged particles

will move in the presence of both

magnetic and electric fields

In that case, the total force is the sum of

the forces due to the individual fields

In general: F = qE + qv x B

Hall Effect (not examinable)

When a current carrying conductor is placed

in a magnetic field, a potential difference is

generated in a direction perpendicular to both

the current and the magnetic field

This phenomena is known as the Hall effect

It arises from the deflection of charge carriers

to one side of the conductor as a result of the

magnetic forces they experience

Hall Voltage

Observing the Hall

effect

The Hall voltage is

measured between

points a and c

Hall Effect

When the charge carriers are negative, the upper edge of the conductor becomes negatively charged

When the charge carriers are positive, the upper edge becomes positively charged

Sign of Hall voltage, VH, gives the sign of the charges

Hall Voltage

In equilibrium qEH = qvdB

VH = EHd = vd B d

d is the width of the conductor

vd is the drift velocity So can be found if B and d are known and VH measured

Where RH = 1 / nq is called the Hall coefficient and y is the

thickness of the conductor, with area A = yd

A properly calibrated conductor can be used to measure the magnitude of an unknown magnetic field

Since vd

I

nqA, then V

HIBd

nqA

IB

nqyRHIB

y

Quick Quiz 29.8

A charged particle is moving perpendicular to a magnetic

field in a circle with a radius r. An identical particle enters

the field, with v perpendicular to B, but with a higher speed

v than the first particle. Compared to the radius of the circle

for the first particle, the radius of the circle for the second

particle is

(a) smaller

(b) larger

(c) equal in size

Answer: (b). The magnetic force on the particle increases in

proportion to v, but the centripetal acceleration increases

according to the square of v. The result is a larger radius, as

we can see from r = mv/qB.

Quick Quiz 29.8

Quick Quiz 29.9

A charged particle is moving perpendicular to a magnetic

field in a circle with a radius r. The magnitude of the

magnetic field is increased. Compared to the initial radius of

the circular path, the radius of the new path is

(a) smaller

(b) larger

(c) equal in size

Answer: (a). The magnetic force on the particle increases in

proportion to B. The result is a smaller radius, as we also see

from r = mv/qB.

Quick Quiz 29.9

End of Chapter

Quick Quiz 29.6

Rank the magnitudes of the torques acting on the rectangular

loops shown in the figure below, from highest to lowest. (All the

loops are identical and carry the same current.)

(a) a, b, c (b) b, c, a (c) c, b, a

(d) a, c, b. (e) All loops experience zero torque.

Answer: (c). Because all loops enclose the same area and

carry the same current, the magnitude of μ is the same for

all. For part (c) in the image, μ points upward and is

perpendicular to the magnetic field and τ = μB, the

maximum torque possible. For the loop in (a), μ points along

the direction of B and the torque is zero. For (b), the torque

is intermediate between zero and the maximum value.

Quick Quiz 29.6

Quick Quiz 29.7

Rank the magnitudes of the net forces acting on the rectangular

loops shown in this figure, from highest to lowest. (All the loops

are identical and carry the same current.)

(a) a, b, c (b) b, c, a (c) c, b, a

(d) b, a, c (e) All loops experience zero net force.

Answer: (e). Because the magnetic field is uniform, there is

zero net force on all three loops.

Quick Quiz 29.7

Quick Quiz 29.10a

Three types of particles enter a mass spectrometer like the one shown

in your book as Figure 29.24. The figure below shows where the

particles strike the detector array. Rank the particles that arrive at a,

b, and c by speed.

(a) a, b, c (b) b, c, a

(c) c, b, a (d) All their speeds are equal.

Answer: (d). The velocity selector ensures that all three

types of particles have the same speed.

Quick Quiz 29.10a

Quick Quiz 29.10b

Rank the particles that arrive at a, b, and c by m/q ratio.

(a) a, b, c (b) b, c, a

(c) c, b, a (d) All their m/q ratios are equal.

Answer: (c). We cannot determine individual masses or

charges, but we can rank the particles by m/q ratio. Equation

29.18 indicates that those particles traveling through the

circle of greatest radius have the greatest m/q ratio.

Quick Quiz 29.10b