# Magnetic Forces and Fields (Chapters...

### Transcript of Magnetic Forces and Fields (Chapters...

Magnetic Forces and Fields

(Chapters 29-30)

• Magnetism – Magnetic Materials and Sources

• Magnetic Field, B

• Magnetic Force

• Force on Moving Electric Charges – Lorentz Force

• Force on Current Carrying Wires

• Applications

• Electromagnetic motors

• Torques on magnetic dipole moments μ

• Sources of Magnetic Field

• Magnetic field of a moving charge

• Current Carrying Wires – Biot-Savart law

• Loops, Coils and Solenoids – Ampère’s Law

• Microscopic Nature of Magnetism

How can one magnetize objects?

• Magnetism can be induced: either by stroking an unmagnetized piece of

magnetizable material with a magnet, or by placing it near a strong permanent magnet

• Soft magnetic materials, such as iron, are easily magnetized

- They also tend to lose their magnetism easily

• Hard magnetic materials, such as cobalt and nickel, are difficult to magnetize

- They tend to retain their magnetism

• Magnets are objects exhibiting magnetic behavior or magnetism

• A magnet exhibits the strongest magnetism at extremities called

magnetic poles: any magnet has two poles, conventionally

dubbed north and south

• Like poles repel each other and unlike poles attract each other

• Unlike charges, magnetic poles cannot be isolated into

monopoles: if a permanent magnetic is cut in half repeatedly, the

parts will still have a north and a south pole

• The region of space surrounding a moving charge includes a magnetic field as well

as by an electric field: so, magnetism and electricity cannot be separated

• They are interrelated into the integrated field of electromagnetism: the first

breakthrough in the great effort of developing unified theories about fundamental

interactions (Maxwell, beginning of the XIX-th century).

Magnetism – Magnets and relation to electricity

Magnetic Field – Operational definition and field lines

• Like the sources of electric field, any magnetic material produces a magnetic field

that surrounds it and extends to infinity.

• The symbol used to represent this vector is

• Let’s first describe this vector using an operational definition:

B

Def: The magnetic field in each location in the surroundings of a magnetic source

is a vector with the direction given by the direction of the north pole of a compass

needle placed in the respective location

• Similar to the electric field, a magnetic field can be patterned using field lines: the

vector B in a point is tangent to the line passing through that point, and the density of

lines represents the strength of the field

• However, while electric field lines start and end on

electric charges (electric monopoles), the magnetic

field lines form closed loops (since there are no

magnetic monopoles)

• Thus, the magnetic field lines should be seen as

closing the loops through the body of the magnet:

that is, the magnetic field inside magnets is not zero

Ex: A compass can be used to

trace the magnetic field lines

• A compass can be used to probe the magnetic field lines produced by various

source, and they will always form closed loops

• Later we’ll look at these sources more systematically:

Magnetic Field – Magnetic field lines for various magnetic sources

Notice the similar

pattern

Magnetic Field – Example: Earth’s Magnetic Field

• The Earth’s geographic north pole is closed to

a (slowly migrating) magnetic south pole

• The Earth’s magnetic field resembles that of a

huge bar magnet deep in the Earth’s interior

slightly tilted with respect to the axis of

rotation of the planet

• The mechanism of Earth’s magnetism is not

very well understood

• There cannot be large masses of permanently

magnetized materials since the high

temperatures of the core prevent materials from

retaining permanent magnetization

• The most likely source is believed to be electric currents in the liquid part of the

planetary core

• The direction of the Earth’s magnetic field reverses every few million years

• The origin of the reversals is not well understood in detail, albeit there are models

describing how it may happen

Magnetic Force – On a moving charge

• Magnetic fields act on moving charges with magnetic forces. We’ll study this effect

in two (related) cases:

1. Moving Charged Particles

2. Current Carrying Wires

1. Magnetic Force on a Moving Point Charge

• Consider a test charge q moving in a field B with velocity

v making an angle θ with B: the particle will be acted by a

magnetic force F (sometimes called Lorentz force) of

Magnitude:

Direction: given by a right hand rule (let’s call it #1):

F

v

B

θ q>0

F q>0

F q<0

sinBF vq

vF Bq

• The expression for the magnetic force leads to a

definition for magnetic field unit called Tesla (T)

• A popular alternative is the cgs unit, Gauss (G) (useful

for small fields): 1T = 104 Gauss

SI

NTesla (T)

C m sB

v

B

Notation: Vectors perpendicular on page/board/slide:

Outward Inward

Exercises:

1. Force direction: Find the direction of the force on an electron moving through the magnetic

fields represented below.

Problem:

1. Charge moving in a magnetic field: What velocity would a proton need to circle Earth 800

km above the magnetic equator, where Earth's magnetic field is directed horizontally north and

has a magnitude of 4.0010-8 T?

2. Field direction: Find the direction of the magnetic

field acting on a proton moving as represented by the

adjacent velocity and force vectors. (Assume that the

velocity is perpendicular on the magnetic field.)

• Any moving charge not only that is acted by a magnetic field but it also produces a

magnetic field that surrounds it and extends to infinity

• A test charge q moving in an electric field E and a magnetic field B, with velocity

making an angle θ with B will be acted by a net electromagnetic force (sometimes

called Lorentz force):

Magnetic Force – Charge in an electromagnetic field

electric magneticF BF q vEF

parallel to the

direction of E

perpendicular on

the direction of B

Ex: One type of velocity selector

• Consider an electric field perpendicular on a magnetic field

• Then only the particles entering the fields with velocity perpendicular

of both will be allowed to pass, which corresponds to the following

condition that the particles are supposed to obey:

+

+

0E

qE qvB vB

Magnetic Force – Trajectory of a point charge in a magnetic field

• Let’s look at two particular trajectories that a charged

particle may have in a magnetic field

1. Consider a particle moving into an external magnetic field

so that its velocity is perpendicular to the field

• In this case, the particle will move in a circle, with the

magnetic force always directed toward the center of the

circular path

• Equating the magnetic and centripetal forces, we can find the radius of the circle:

2vF qvB m

r

mr

B

v

q

+

+

+

: called cyclotron equation

2. If the particle’s velocity is not perpendicular to

the field, the path followed by the particle is a spiral

called a helix

• The helix spirals along the direction of the field

with a velocity given by the component of the

velocity parallel with B

v

v

v

vB

F

FF

+

• A current is a collection of many drifting charged particles, such that a magnetic

force is expected to act on a current-carrying wire placed in a magnetic field

• This magnetic force is the resultant of the forces acted on the individual microscopic

electric carriers, but it makes more sense to integrate its effects into a unique

magnetic force acted on the macroscopic current

I = 0 F = 0 I ↑ F ← I ↓ F→

Magnetic Force – Currents in magnetic field

Ex: Experimental observations:

A current carrying vertical wire

placed in a magnetic field pointing

perpendicular into the slide, will be

acted by a magnetic force

perpendicular on the current and

magnetic field: either to the left, or

to the right, depending on the

direction of the current

2. Magnetic Force on Current Carrying Wire

• Consider a straight current carrying wire of length ℓ

immersed in field B, making an angle θ with B: the

portion dℓ of wire will be acted by a magnetic force dF

Magnitude:

Direction: Given by right hand rule #1, but instead of

aligning the fingers with the velocity, one aligns the

fingers with the direction of the current

• Since the current flows in the direction of the

positive carriers, the thumbs always indicates the

direction of the force

• If the wire is straight, and the force is the same for

each cross-section, the force on a length L of wire is

F

B

θ

F

I

sindF dBI

Magnetic Force – On a current carrying wire

dF d BI

I

d

sinF LIB B

Problems:

2. Basics of a rail gun: A rail gun looks (very)schematically

as in the figure. Evaluate the speed that the projectile of

mass m would achieve after traveling from rest a distance d

on the rails spaced by ℓ with a driving current I with a

magnetic field B.

3. Force on a semicircular current: A semicircular thin

conductor of radius R carries a time dependent current

,

where I0 and τ are positive constant. The wire is allowed to

move vertically through a uniform magnetic field B, as in the

figure. Find the acceleration of the conductor as a time

dependent function.

0

ti I e

i

I

BI

Projectile

Rail

ℓ

m

𝐵

R

y

x

θ

R

Applications – Torque on a Current Loop

• The magnetic force can be used to make electromagnetic motors by using it to rotate

a current carrying loops

1 12 2

sin sin sinF a F a Fa

F BIb

• We see that the torque is maximum when the magnetic field B is parallel with the

plane of area A (θ = 90°), and zero when B is perpendicular on A (θ = 0°)

• In order to see the principles of such an arrangement, consider

a loop carrying a current I in an external magnetic field B

• The two sides perpendicular on B will be acted by forces

opposite in direction creating a torque that will rotate the loop:

F F

sin sinBIba BIA

• We can immediately find an expression for the net torque τ = τ1 + τ2:

11 1 2

sinF a

12 2 2

sinF a

angle between the magnetic

field and the perpendicular to

the surface of the loop

I I

I I

I

I

Magnetic Moment, μ

• The net torque exerted by a magnetic field on N current carrying loops is

sinNIAB

IA

Ex: Electrons in an atom have an orbital moment due to the

their orbital motion about the nucleus

• Any loop of electric current can be associated with a

magnetic moment pointing perpendicular on the plane

of the loop

• So, we see that a magnetic dipole in a magnetic field

will have the tendency to rotate either in a position with

μ parallel with B – stable equilibrium – or anti-parallel

with B – unstable equilibrium

• The “current” doesn’t have to be carried by a wire:

any closed loop of moving charges will have a moment:

as we shall see later, these moments explain magnetism

at a microscopic scale

• This magnetic torque exerted on the loop of current can be

written in terms of a vector quantity called magnetic moment:

sinB B

+

Applications – Electromagnetic Motors

• An electric motor converts electrical energy into mechanical energy in the form of

rotational kinetic energy

• As described on the previous slides, the simplest electric motor consists of a rigid

current-carrying loop that rotates when placed in a magnetic field

• The torque acting on the loop will tend to rotate

the loop to smaller values of θ until the torque

becomes 0 at θ = 0°

• If the loop turns past this point and the current

remains in the same direction, the torque reverses

and turns the loop in the opposite direction

• To provide continuous rotation in one direction,

the current in the loop must periodically reverse,

such that dc-motors must use split-ring

commutators and brushes

• Actual motors would contain many current loops

and commutators

Magnitude:

Direction: perpendicular on the plane determined by

r and v. Use the following right hand rule (#2): grab

the velocity in your right hand with the thumb in its

direction. Then the curl of the fingers will show the

direction of the field around v: clockwise for positive

charges and anticlockwise for negative charges

Sources of Magnetic Field – Moving Charge

• Consider a point charge moving with constant velocity

v: then, the magnetic field B at a position r from the

particle making an angle θ with v is

where µ0= 4 10-7 Tm/A is the magnetic permeability

of free space

0

2

ˆ

4

q rvB

r

v

0

2

sin

4

vqB

r

θ

v

• We’ve seen that magnetic fields act on moving charges (point-like and currents), so

it is just natural to expect that moving charges also produce magnetic fields: a fact

first discovered serendipitously by Hans Oersted in 1819

r

B

+

Larger field in

the plane

Weaker field

behind

Weaker fields

ahead

B

+

BB

• Consider a current I carried along a wire. Then, the

magnetic field produced by a segment dℓ of the current

at a position r from the segment making an angle φ

with dℓ is given by

0

2

ˆ

4

IddB

r

r

0

2

sin

4

IddB

r

0

24

ˆId rB

r

I

Sources of Magnetic Field – Element of current

φ

r

dB

I

d

Biot-Savart Law:

Magnitude:

Direction: perpendicular on the plane determined by r and

v. Use the same right hand rule as for moving positively

charged particles, but curl your right hand fingers around

the current.

• Hence, for a certain finite length of wire

dB

dB

dB

dB

Problem:

4. Moving charges interacting electrically and magnetically: Two protons move with

uniform speed v along parallel paths at distance r from each other.

a) Find a symbolical expression for the magnetic force exerted by one proton on the other

one: is it attractive or repulsive? Is this always the case?

b) Calculate the electric force between the charges and compare to the magnetic force.

+

+

Problems:

5. Straight Current: A straight wire of length 2a centered in y = 0, carries a current I in

positive y-direction. Calculate the magnetic field at distance r along x-axis.

Useful integral:

6. Circular Current: A circular wire loop of radius a lays in the yz-plane and is traveled by a

counterclockwise current I.

a) Calculate the magnetic field produced at a distance x along the axis of the loop.

b) Using the result, find the field in the center of the ring.

3 2 1 2 1 2

2 2 2 2 2 2

2

a

a

aa

dy y a

x y x x y x x a

Sources of Magnetic Fields – Long straight wire

• Consider a long straight wire carrying a current I, the magnetic field at a distance r

perpendicular on the wire is given by:

Magnitude: using the result for Problem 5:

Direction: Given by the right hand rule #2

Comments:

• The magnetic field has cylindrical

symmetry around the wire

• It gets weaker and weaker as the

circles are larger and larger

0

2B

r

I

0

2 2

1

2 1a

IB

r r a

a) Half as strong, same direction.

b) Half as strong, opposite direction.

c) One-quarter as strong, same direction.

d) One-quarter as strong, opposite direction.

Quiz 1: A long wire carries a current I as in the figure. Compared to

the magnetic field at point A, the magnetic field at point B is

I

I

Magnetic Force Between Two Parallel Conductors

• If two long current carrying wires are placed parallel with

each other, they will interact via magnetic forces

• The force on wire 1 is due to magnetic field produced by

wire 2 onto the current in 1, so the force per unit length L

is:

Comments:

• Parallel currents attract each other whereas anti-parallel

conductors repel each other

0

2

F II

L r

• The force between parallel conductors can be used to redefine the Ampere (A)

• Then the Coulomb (C) can be also defined in terms of the Ampere

0

2

IB

rF I L I L

Def: If a conductor carries a steady current of 1 A, then the quantity of charge

that flows through any cross section in 1 second is 1 C

Def: If two long, parallel wires 1 m apart carry the same current, and the

magnitude of the magnetic force per unit length is 2 x 10-7 N/m, then the current

is defined to be 1 A

Problems:

7. Force on a moving particle by a current carrying wire: A

proton moves with speed v = 0.25 m/s parallel with a long

wire carrying a current I = 2.0 A, at distance r = 1.0 mm.

Calculate the magnetic force on the proton.

8. Superposition of aligned magnetic fields: Two long

parallel wires carry currents I1 and I2 in opposite directions.

The figure is an end view of the conductors. Calculate the

magnitudes of the magnetic field in points A, B and C located

at given equal distances a from the closest wires.

a a a a

a b c

I1 I2

r + I

v

e+

N

S

BB

I

• If we allow x → 0 in the result of Problem 6, the magnetic field in the center of a

circular loop is

• N loops form a coil

with the maximum field

in the middle of the coil:

2

0

3 2 02 22 x

I aB

x a

Sources of Magnetic Fields – Circular loop of current. Coils

The magnetic field B inside

has the same direction as

the magnetic moment μ

I • Notice that a current loop can be seen as a

magnet with magnetic field lines that remind

of the equipotential lines of an electric dipole:

so the loop behaves like a magnetic dipole

• The field produced by a loop or a coil is related to the

respective magnetic moment μ

Ex: Magnetic field and moment: the magnetic field in the

center of a circular loop is related to its magnetic moment

as given by

0

2B

a

I

2 0

32Ba I

a

0max

2B N

a

I

249.27 10 J TB

Magnetism in Materials – Magnetic moments of electrons

• Now we are prepared to see that the magnetism of materials is microscopically

mainly determined by the alignment of elementary electronic magnetic dipoles

• Notice first that atoms should act like magnets because of

the orbital motion of the electrons about the nucleus

• Since each electron circles the atom once in about every

10-16 seconds, it produces a current of 1.6 mA and a

magnetic field of about 20 T at the center of the orbit

• However, the magnetic field produced by one electron in

an atom is often canceled by an oppositely revolving

electron in the same atom, so the net result is that the

magnetic effect produced by electrons orbiting the nucleus

is either zero or very small for most materials

• The classical model is to consider the electrons to spin like tops but

it is actually a quantum effect.

• The magnetic moment of an electron is given by Bohr magneton:

• Most materials are not naturally magnetic since electrons usually pair up with their

spins opposite each other

+

spin

Magnetism in Materials – Types of magnetism

Paramagnets Ex: aluminum, uranium

• Moments point in random directions in zero external field but

in an external field Bext they rotate so the net field increases

• The increment in field is small and paramagnetism competes

with thermal motion

Ferromagnets Ex: iron, nickel

• In some materials, large groups of atoms in which the spins are

aligned form ferromagnetic domains

• When an external field Bext is applied, the domains that are

aligned with the field tend to grow at the expense of the others

• This causes the material to become magnetized by amounts

larger than in the paramagnetic case

Diamagnets Ex: mercury, superconductors, animal bodies

• In these materials an external magnetic field induces opposite magnetic

• In these cases the internal magnetic field is less than the external one

• A diamagnet placed in an external magnetic field will have the tendency to float

• The magnetization is slowly disappearing after removing the external field

• So, since the alignment of elementary magnetic dipoles associated with the electronic spins

depends on the microscopic structure, various materials are classified depending on how they

behave in external magnetic fields:

extB

extB

B

I1 I2

I3

I4

Iencl = I2 + I3 + I4

d

B

B

Ampère’s Law

• André-Marie Ampère found a procedure for deriving the relationship between the

current in an arbitrarily shaped wire and the magnetic field produced by the wire:

Net current inside

the path

Ampère’s Circuital Law: If a net current Iencl is

enclosed by an arbitrary closed path, the integral of all

products B|| dℓ (where B|| is the component of the

magnetic field along each elementary step dℓ of the

path) is proportional to Iencl:

Line integral around a closed

path called an Amperian loop

Ex: Ampere’s law can be used to demonstrate the result that we

obtained previously for a closed circular path around a long

straight current I: since the circumference of the path is 2 r,

and by symmetry the field around the Amperian is expected to

be everywhere constant and tangent to the path, we get:

002

2encl

IB d B d B r I B

r

0 enclB d I

Amperian

loop

I

B r

Problem:

8. Magnetic field inside and outside of a current carrying conductor: A cylindrical

conductor with radius R carries a current I uniformly distributed over the cross-sectional

area of the conductor. Confirm the relationships given below for the magnetic field in the

interior and the exterior of the conductor.

I

Direction

• The field lines of the solenoid resemble those of a bar

magnet and the field direction is given by the right hand

rule applied to the current through any of the turns

Magnitude

• The magnitude of the field inside a solenoid is constant at

all points far from its ends

where n is the number of turns per unit length

• This expression can be obtained by applying Ampère’s

Law to the solenoid…

0B n I

n N

Quiz 2: What is the

direction of the field in the

center of the solenoid

below?

Sources of Magnetic Field – Solenoids

• If a long straight wire is wound into

a coil of closely spaced loops, the

resulting device is called a solenoid

• It is also known as an electromagnet

since it acts like a magnet only when

it carries a current

S

S

N

N

I

I

B

solenoid bar magnet

I

Sources of Magnetic Field – Solenoid field using Ampère’s Law

• Consider a cross-sectional view of a tightly wound solenoid of turn density n,

carrying a current I

• If the solenoid is long compared to its radius, we assume the field inside is uniform

and outside is zero

• Then we can apply Ampère’s Law to a rectangular Amperian a → b → c → d → a:

0 0 0 N

BL NI B I n IL

Comment:

In reality the field is not perfectly

uniform along the axis of a cylinder. Amperian loop

with N turns inside

Ampère’s law

I