Chapter 20 - Magnetic Field

practical devices depends on

magnetism, from compasses to motors, loudspeakers, computer memory, and electric generators 

magnetic field are created by

• the earth

• permanent magnets

• electric current as described by Ampere’s law

we have all played with magnets at some time, you may have some stuck to your fridge right now now; these types of magnets are known as 

permanent magnets 

permanent magnets are made from 

ferromagnetic materials 

typeical ferromagnetic metals are

iron, cobalt, and nicket

• rocks containing these elements are magnetized when they form due to the Earth’s magnetic field 

a standard bar magnet has two ends, known as a

North Pole (N) and a South Pole (S)

two bar magnets interact through

magnetic force

• N repels N, S repels S

• N attracts S, S attracts N

a bar magnet produces a

magnetic field,

• and a bar magnet responds to a magnetic field 

if you cut a bar magnet in half, you do not get 

separate north and south poles

• you get two, smaller bar magnets, each with a north and South Pole 

a single pole (north or south) is known as a

magnetic monopole

• magnetic monopoles do not exist 

a North Pole and a South Pole together, is known as a

magnetic pole (most basic magnetic object)

the magnetic field of a bar magnetic is the field of

a dipole

a circular loop of current produces a

magnetic field that is identical to a bar magnetic (in an ideal sense)

many fundamental particles and some atoms and molecule also produce

dipole magnetic fields

how these tiny dipoles respond to an external magnetic field determines the type of magnetic material:

ferromagnetic, paramagnetic, diamagnetic 

in ferromagnetic material, there are domains of the tiny dipoles that are aligned in 

the same direction, even when there is no external field `

in paramagnets and diamagnets, the tiny dipole align when there is an 

external field, but they are not aligned otherwise 

this is some sense is the root of all magnetic fields and the reason that

magnetic monopoles do not exist

introducing the magnetic field lines - we used

small, positive charges to map out the electric field lines

the direction of an electric field line at some point is the direction a

small, positive charge would want to move when placed at that point

for magnetic field lines, we will use small bar magnets - the direction of the magnetic field line at some point is the direction that

the North Pole of a small bar magnet would point when placed at that point

place a small bar magnet near the north pole of a large bar magnet

• N repels N, so the small bar magnet will want to turn so that its N points away

• therefore, the magnetic fields of the large bar magnet, near its North Pole, point towards it 

place a small bar magnet near the south pole of the large bar magnet

• S attracts N, so the small bar magnet will want to turn so that its N points towards

• therefore, the magnetic field lines of the large bar magnet, nears its South Pole, point towards it

it turns out since there are no magnetic monopoles, magnetic field lines must

form closed loops

if a N monopole existed, field lines would

originate on it (like a positive point charge)

if a S monopole existed, field lines would

terminate on it (like a negative point charge)

magnetic field lines - the magnetic field at a point is 

tangent (same slope) to magnetic field lines near the point, and points in the direction of the field lines

magnetic field lines - where the magnetic field is stronger, the field lines will

be closer together (denser)

magnetic field lines - where the magnetic field is weaker, the field lines will be 

further apart (sparser) 

magnetic field lines - field lines cannot

cross (otherwise there would be two tangent directions at the same point)

magnetic field lines - plus, they form

closed loops

1. the lines emerge form the North Pole

2. lines enter at the South Pole 

since the North Pole of a compass needle points roughly north, the south magnetic pole must close to the

geographic North Pole 

the position of the magnetic poles do not 

coincide with the position of geographic poles

• there is geographic north (true north) and magnetic north

  • the difference between the two is known as the magnetic declination, and varies place to place

at most locations, the field is not

parallel to surface, but has a Dip Angle

a compass needle is simply a

bar magnet which is supported at its center of gravity so that it can rotate freely

Ampere’s law relates the magnetic field around a loop to the current that crosses the loop

• the loop does not have to be physical 

• the loop is known as an Amperian loop (like a gaussian surface) 

the sum of the magnetic field around a closed loop is

proportional to the net current passing through the loop

currents contribute to the net enclosed current based on

direction relative to the direction of the sum around the loop

long, straight wire has cylindrical symmetry - the magnetic field produced by a current in such a wire can only depend on

perpendicular distance from the wire 

long, straight wire has cylindrical symmetry - therefore, the magnitude of the magnetic field at distance r from the wire is 

• inversely proportional to the distance from the wire

• proportional to the current 

when an arrow is coming towards you, you see the point, so

a dot will represent a vector that points out of the page

when an arrow is headed away from you, you see the tail feathers, so

a cross will represent a vector that points into the page

the magnetic field lines are

circles, centered on the wire

their direction is given by the right-hand rule: thumb of right hand 

in direction of current 

their direction is given by the right-hand rule: fingers of right-hand

curl in the direction of the field lines

recall that the field at some point is always

tangent to a field line near that point

• this means that the field never points towards the wire and never points away from it 

superposition Principe applies: net field is the vector sum of the

individual fields of the currents

we can picture the magnetic field of a circular loop by imagining taking

a straight wire and bending it into a circle

we also argued previously that the field of a current loop is a

magnetic dipole

a solenoid is like a

long coil, or spring

• we can think of it as a bunch of stacked coils

if the solenoid is much longer than it is wide, the field in the interior, away from the ends, will be approximately

uniform, and point along the axis of the solenoid, as given by the right hand rule like a current loop

an electric motor changes electric energy into

(rotational) mechanical energy

principle: a torque is exterted on a

current-carrying loop in a magnetic field

a current is moving charges; therefore, a magnetic field will exert a 

force on a current carrying wire 

current-carrying wire- this a cross product, which means that

the force is perpendicular to both the current and the field

consider a rectangular loop in a uniform magnetic field, the net force on the loop

is zero

in fact, the net force on any current loop in a uniform magnetic field is

zero

there can be a net force in a 

non-uniform field 

if wire 1 exerts a force on wire 2, then by newton’s 3rd law, wire 2 must

exert an equal and opposite force on wire 1

the force is attractive when the two currents are

in the same direction

the force is repulsive for two currents in

opposite direction 

the magnetic field can exert a force on a

moving electric charge

• depends on charge, velocity, magnetic field strength, and the angle between the velocity and the magnetic field

a magnetic field will not exert a force on an electric charge unless

it is moving

the direction of the force on a positive charge with velocity v in a magnetic field B is given by a 

right-hand rule (RHR)

using your right hand (Version 1)

• orient your right hand so that your outstretched fingers
  point along the direction of the velocity

• bend your fingers so that they must point along the
  direction of the field

• Your thumb will point in the direction of the force

if the charge is negative, the force will be in the

direction opposite of where your thumb is pointing

using your right hand (version 2)

• Put your index finger straight out and in the direction
  of the velocity

• Put your middle finger in the direction of the field

• Make sure your thumb is perpendicular to both fingers

• Your thumb will point in the direction of the force

if a charge is releases from rest in an electric field, it experiences an

electric force, leading to a constant acceleration

if a charge is at rest in a magnetic field, there won’t be

a magnetic force acting on it, as its velocity is zero

the magnetic force is always perpendicular to

the velocity of the particle (displacement)

the magnetic force never

does work

since work done by magnetic field is zero, this also means there is no

potential energy (PE) associated with the magnetic force, and therefore no magnetic potential either

kinetic energy is also not change and therefore

speed is constant

uniform magnetic field - consider an electron at point P, moving to the right

• the force on the electron at the point will point down

• therefore the electron will curve downward

uniform magnetic field -moving from P to Q, the direction of the velocity turns, and direction of the force does

too (must be perpendicular to velocity)

uniform magnetic field - the path is a circle with the force always

pointing towards the center

uniform magnetic field - for a charge in a magnetic field, the force and therefore acceleration is 

always perpendicular to the velocity 

• the acceleration will be perpendicular to its velocity 

• whenever this is tire, the speed of the object is constant, only direction changes 

if the field is uniform, then the acceleration will have a

constant magnitude

conductor in magnetic field - the conduction electron will therefore migrate to one end of the rod

• that end will start to develop a net negative charge, the other end a
  net positive charge

• This will induce an electric field (E) within the rod, pointing from
  positive to negative

• This will create an electric force (𝑭e) on the conduction, pointing in
  the opposite direction as the magnetic force, and with a magnitude

conductor in magnetic field - the electric force will grow until it balances the magnetic force, at which point the net force

on conduction electrons will be zero

a changing magnetic flux induces an

electromotive force (emf)

in a closed loop of wire, there will also be an induced current, which implies there is an 

electric field in the wire causing the electrons to start moving 

in general: a changing magnetic flux produces

an electric field