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

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 4.0010-8 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:

+

+

0 E

qE qvB v B

    

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:

2v F qvB m

r   

m r

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

v B

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 cu