Magnetic fields

The symbol we use for a magnetic field is B.

The unit is the tesla (T).

The Earth’s magnetic field is about 5 x 10-5 T.

Which pole of a magnet attracts the north pole of a compass? Which way does a compass point on the Earth?

What kind of magnetic pole is near the Earth’s geographic north pole?

What are some similarities between electric and magnetic fields? What are some differences?

Similarities between electric and magnetic fields

• Electric fields are produced by two kinds of charges, positive and negative. Magnetic fields are associated with two magnetic poles, north and south, although they are also produced by charges (but moving charges).

• Like poles repel; unlike poles attract.

• Electric field points in the direction of the force experienced by a positive charge. Magnetic field points in the direction of the force experienced by a north pole.

Differences between electric and magnetic fields

• Positive and negative charges can exist separately. North and south poles always come together. Single magnetic poles, known as magnetic monopoles, have been proposed theoretically, but a magnetic monopole has never been observed.

• Electric field lines have definite starting and ending points. Magnetic field lines are continuous loops. Outside a magnet the field is directed from the north pole to the south pole. Inside a magnet the field runs from south to north.

Observing a charge in a magnetic field

The force exerted on a charge in an electric field is given by

Is there an equivalent equation for the force exerted on a

charge in a magnetic field? Simulation

Case 1: The charge is initially stationary in the field.

Case 2: The velocity of the charge is parallel to the field.

F qE

Observing a charge in a magnetic field

The force exerted on a charge in an electric field is given by

Is there an equivalent equation for the force exerted on a

charge in a magnetic field? Simulation

Case 1: The charge is initially stationary in the field.

The charge feels no force.

Case 2: The velocity of the charge is parallel to the field.

The charge feels no force.

F qE

Observing a charge in a magnetic field

Simulation

Case 3: Three objects, one +, one -, and one neutral, have an initial velocity perpendicular to the field. The field is directed out of the screen.

Observing a charge in a magnetic field

Simulation

Case 3: Three objects, one +, one -, and one neutral, have an initial velocity perpendicular to the field. The field is directed out of the screen.

Magnetic fields exert no force on neutral particles.

The force exerted on a + charge is opposite to that exerted on a – charge.

The force on a charged particle is perpendicular to the velocity and the field. In this special case where the velocity and field are perpendicular to one another, we get uniform circular motion.

Observing a charge in a magnetic field

Simulation

Case 4: The same as case 3, except the magnetic field is doubled.

Observing a charge in a magnetic field

Simulation

Case 4: The same as case 3, except the magnetic field is doubled.

We observe the radius of the path to be half as large.

Thus, doubling the magnetic field doubles the force -

the force is proportional to the magnetic field.

2mvF

r

Observing a charge in a magnetic field

Simulation

Case 5: Three positive charges +q, +2q, and +3q are initially moving perpendicular to the field with the same velocity.

Observing a charge in a magnetic field

Simulation

Case 5: Three positive charges +q, +2q, and +3q are initially moving perpendicular to the field with the same velocity.

We observe the radius of the path to vary inversely with the charge.

Thus, doubling the charge doubles the force, and tripling the charge triples the force -

the force is proportional to the charge.

2mvF

r

Observing a charge in a magnetic field

Simulation

Case 6: Three identical charges, are initially moving perpendicular to the field with initial velocities of v, 2v, and 3v, respectively.

Observing a charge in a magnetic field

Simulation

Case 6: Three identical charges, are initially moving perpendicular to the field with initial velocities of v, 2v, and 3v, respectively. We observe the radius of the path to be proportional to the speed. However:

What does this tell us about how the force depends on speed?

2mvr

F

Observing a charge in a magnetic field

Simulation

Case 6: Three identical charges, are initially moving perpendicular to the field with initial velocities of v, 2v, and 3v, respectively. We observe the radius of the path to be proportional to the speed. However:

What does this tell us about how the force depends on speed? The force is proportional to the speed.

2mvr

F

Summarizing the observations• There is no force applied on a stationary charge by a magnetic field, or on a charge moving parallel to the field.

• Reversing the sign of the charge reverses the direction of the force.

• The force is proportional to q (charge), to B (field), and to v (speed).

Summarizing the observations• There is no force applied on a stationary charge by a magnetic field, or on a charge moving parallel to the field. • Reversing the sign of the charge reverses the direction of the force. • The force is proportional to q (charge), to B (field), and to v (speed).

The magnitude of the force is F = q v B sin(θ), where θ is the angle between the velocity vector v and the magnetic field B.

The direction of the force, which is perpendicular to both v and B, is given by the right-hand rule.

Something to keep in mindA force perpendicular to the velocity, such as the magnetic force, can not change an object’s speed (or the kinetic energy). All it can do is make the object change direction.

The right-hand rulePoint the fingers of your right hand in the direction of the velocity.

Curl your fingers into the direction of the magnetic field (if v and B are perpendicular, pointing your palm in the direction of the field will orient your hand properly).

Stick out your thumb, and your thumb points in the direction of the force experienced by a positive charge. If the charge is negative your right-hand lies to you. In that case, the force is opposite to what your thumb says.

Simulation

The right-hand rule

Practice with the right-hand rule

In what direction is the force on a positive charge with a velocity to the left in a uniform magnetic field directed down and to the left?

1. up

2. down

3. left

4. right

5. into the screen

6. out of the screen

7. a combination of two of the above

8. the force is zero

9. this case is ambiguous - we can't say for certain

Practice with the right-hand rule

v and B define a plane, and the force is perpendicular to that plane. The right-hand rule tells us the force is out of the screen. We use a dot symbol to represent out of the screen (or page), and an x symbol to represent into the screen.

Practice with the right-hand rule, II

In what direction is the force on a negative charge, with a velocity down, in a uniform magnetic field directed

out of the screen? 1. up

2. down

3. left

4. right

5. into the screen

6. out of the screen

7. a combination of two of the above

8. the force is zero

9. this case is ambiguous - we can't say for certain

Practice with the right-hand rule, II

Remember that with a negative charge, your right hand lies to you – take the opposite direction.

Practice with the right-hand rule, III

In what direction is the force on a positive charge that is initially stationary in a uniform magnetic field directed into the screen?

1. up

2. down

3. left

4. right

5. into the screen

6. out of the screen

7. a combination of two of the above

8. the force is zero

9. this case is ambiguous - we can't say for certain

Practice with the right-hand rule, III

Magnetic fields exert no force on stationary charges.

Practice with the right-hand rule, IV

In what direction is the force on a negative charge with a velocity to the left in a uniform electric field directed out of the screen?

1. up

2. down

3. left

4. right

5. into the screen

6. out of the screen

7. a combination of two of the above

8. the force is zero

9. this case is ambiguous - we can't say for certain

Practice with the right-hand rule, IV

We don’t need the right-hand rule for an electric field, we need . The force is opposite to the field, for a negative charge.

F qE

Practice with the right-hand rule, V

In what direction is the velocity of a positive charge if it feels a force directed into the screen from a magnetic

field directed right? 1. up

2. down

3. left

4. right

5. into the screen

6. out of the screen

7. a combination of two of the above

8. the force is zero

9. this case is ambiguous - we can't say for certain

Practice with the right-hand rule, V

This is ambiguous. The right-hand rule tells us about the component of the velocity that is perpendicular to the field, but it can’t tell us anything about a component parallel to the field – that component is unaffected by the field.

Charges moving perpendicular to the field

The force exerted on a charge moving in a magnetic field is always perpendicular to both the velocity and the field.

If v is perpendicular to B, the charge follows a circular path.

The radius of the circular path is:

2mvF qvB

r

mvrqB

Charges moving perpendicular to the field

The radius of the circular path is:

The time for the object to go once around the circle (the period, T) is:

Interestingly, the time is independent of the speed. The faster the speed, the larger the radius, but the period is unchanged.

2 2 / 2r mv qB mT

v v qB

mvrqB

Circular paths

Three charged objects with the same mass and the same magnitude charge have initial velocities directed right. Here are the trails they follow through a region of uniform magnetic field. Rank the objects based on their speeds.

1. 1 > 2 > 3

2. 2 > 1 > 3

3. 3 > 2 > 1

4. 3 > 1 > 2

5. None of the above

Circular paths

The radius of the path is proportional to the speed, so the

correct ranking by speed is choice 2, 2 > 1 > 3.

Circular paths, II

Three charged objects with the same mass and the same magnitude charge have initial velocities directed right. Rank the objects based on the magnitude of the force they experience as they travel through the magnetic field.

1. 1 > 2 > 3

2. 2 > 1 > 3

3. 3 > 2 > 1

4. 3 > 1 > 2

5. None of the above

Circular paths, II

The force is proportional to the speed, so the correct ranking

by force is also choice 2, 2 > 1 > 3.

Possible paths of a charge in a magnetic field

If the velocity of a charge is parallel to the magnetic field, the charge moves with constant velocity because there's no net force.

If the velocity is perpendicular to the magnetic field, the path is circular because the force is always perpendicular to the velocity.

What happens when the velocity is not one of these special cases, but has a component parallel to the field and a component perpendicular to the field?

The parallel component produces straight-line motion. The perpendicular component produces circular motion. The net motion is a combination of these, a spiral. Simulation

Which way is the field?

The charge always spirals around the magnetic field. Assuming the charge in this case is positive, which way does the field point in the simulation?

1. Left

2. Right

Spiraling chargesCharges spiral around magnetic field lines.

Charged particles near the Earth are trapped by the Earth’s magnetic field, spiraling around the Earth’s magnetic field down toward the Earth at the magnetic poles.

The energy deposited by such particles gives rise to ??

Spiraling chargesCharges spiral around magnetic field lines.

Charged particles near the Earth are trapped by the Earth’s magnetic field, spiraling around the Earth’s magnetic field down toward the Earth at the magnetic poles.

The energy deposited by such particles gives rise to the aurora borealis (northern lights) and the aurora australis (southern lights). The colors are usually dominated by emissions from oxygen atoms.

Photos from Wikipedia

A mass spectrometerA mass spectrometer is a device for separating particles based on their mass. There are different types – we will investigate one that exploits electric and magnetic fields. Step 1: Accelerate charged particles via an electric field.

Step 2: Use an electric field and a magnetic field to select particles of a particular velocity.

Step 3: Use a magnetic field to separate particles based on mass.

Step 1: The AcceleratorSimulation

The simplest way to accelerate ions is to place them between a set of charged parallel plates. The ions are repelled by one plate and attracted to the other. If we cut a hole in the second plate, the ions emerge with a kinetic energy determined by the potential difference between the plates.

K = | q V |

Step 3: The Mass SeparatorSimulation

In the last stage, the ions enter a region of uniform magnetic field B/. The field is perpendicular to the velocity. Everything is the same for the ions except for mass, so the radius of each circular path depends only on mass.

2mv mvF qvB r

r qB

Step 3: The Mass Separator

The ions are collected after traveling through half-circles, with the separation s between two ions is equal to the difference in the diameters of their respective circles.

2mv mvF qvB r

r qB

2 12 1

2( )2 2

m m vs r r

qB

Magnetic field in the mass separator

In what direction is the magnetic field in the mass separator? The paths shown are for positive charges.

1. up

2. down

3. left

4. right

5. into the screen

6. out of the screen

Step 2: The Velocity SelectorSimulation

To ensure that the ions arriving at step 3 have the same velocity, the ions pass through a velocity selector, a region with uniform electric and magnetic fields.

The electric field comes from a set of parallel plates, and exerts a force of on the ions.

The magnetic field is perpendicular to both the ion velocity and the electric field. The magnetic force, , exactly balances the electric force when:

Ions with a speed of

pass straight through.

EF qE

MF qvB

qE qvB E vB E

vB

Magnetic field in the velocity selector

In what direction is the magnetic field in the velocity selector, if the positive charges pass through undeflected? The electric field is directed down.

1. up

2. down

3. left

4. right

5. into the screen

6. out of the screen

Magnetic field in the velocity selectorThe right-hand rule tells us that the magnetic field is directed into the screen.

Negative ions in the velocity selector

If the charges passing through the velocity selector were negative, what (if anything) would have to be changed for the velocity selector to allow particles of just the right speed to pass through undeflected?

1. reverse the direction of the electric field

2. reverse the direction of the magnetic field

3. reverse the direction of one field or the other

4. reverse the directions of both fields

5. none of the above, it would work fine just the way it is

Negative ions in the velocity selectorIf the charges are negative, both the electric force and the magnetic force reverse direction. The forces still balance, so we don’t have to change a thing.

Faster ions in the velocity selector

Let’s go back to positive ions. If the ions are traveling faster than the ions that pass undeflected through the velocity selector, what happens to them? They get deflected …

1. up

2. down

3. into the screen

4. out of the screen

Faster ions in the velocity selectorFor ions with a larger speed, the magnetic force exceeds the electric force and those ions are deflected up of the beam. The opposite happens for slower ions, so they are deflected down out of the beam.

A cyclotronSimulation

A cyclotron is a particle accelerator that is so compact a small one can fit in your pocket. It consists of two D-shaped regions known as dees. In each dee there is a magnetic field perpendicular to the plane of the page. In the gap separating the dees, there is a uniform electric field pointing from one dee to the other. When a charge is released from rest, it is accelerated by the electric field and carried into a dee. The magnetic field in the dee causes the charge to follow a half-circle that carries it back to the gap.

A cyclotronErnest Lawrence won the 1939 Nobel Prize in Physics for inventing the cyclotron. The cyclotron has many applications, including accelerating ions to high energies for medical treatments. A good example is the Proton Therapy Center at Mass General Hospital (see below). After leaving the cyclotron, the beam is steered using magnetic fields.

Magnetic fields in the dees

In what direction is the magnetic

field in each of the dees? The

path shown is for a positive

charge.

1. out of the screen in both dees

2. into the screen in both dees

3. out of the screen in the left dee; into the screen in the right dee

4. into the screen in the left dee; out of the screen in the

right dee

Increasing the energy

You want to increase the speed of the particles when they emerge from the cyclotron. Which is more effective, increasing the potential difference across the gap or increasing the magnetic field in the dees?

1. increasing the potential difference in the gap

2. increasing the magnetic field in the dees

3. either one, they're equally effective

The energy increases by ΔK each time the charge crosses the gap, and stays constant in the dees.

Producing a magnetic fieldElectric fields are produced by charges.

Magnetic fields are produced by moving charges.

In practice, we generally produce magnetic fields from currents.

The magnetic field from a long straight wire

The long straight current-carrying wire, for magnetism, is analogous to the point charge for electric fields.

The magnetic field a distance r

from a wire with current I is:

, the permeability of free space, is:

0

2

IB

r

0

70 4 10 Tm/A

The magnetic field from a long straight wire

Magnetic field lines from a long straight current-carrying wire are circular loops centered on the wire.

The direction is given by another

right-hand rule.

Point your right thumb in the

direction of the current

(out of the screen in the

diagram, and the fingers on

your right hand, when you curl

them, show the field direction.

The net magnetic field

In which direction is the net magnetic field at the origin in the situation shown below? All the wires are the same distance from the origin.

1. Left

2. Right

3. Up

4. Down

5. Into the page

6. Out of the page

7. The net field is zero

The net magnetic field

We add the individual fields to find the net field, which is directed right.

Four wiresThink about the net magnetic field at the center of the square because of four wires, one at each corner, that carry currents of the same magnitude. Can you choose current directions for the four wires so that the net magnetic field at the center is directed toward the top right corner of the square?

How many ways?

You can choose the direction of the currents at each corner. How many configurations give a net magnetic field at the center that is directed toward the top-right corner?

1. 1

2. 2

3. 3

4. 4

5. 0 or more than 4

How many ways?First, think about the four fields we need to add to get a net field toward the top right. How many ways can we create this set of four fields?

How many ways?How many ways can we create this set of four fields?

Two. Wires 2 and 4 have to

have the currents shown.

Wires 1 and 3 have to

match, so their fields cancel.

The right-hand rule:

Point your thumb in the

direction of the current,

and your curled fingers

show the direction of the field.

The force on a current-carrying wireA magnetic field exerts a force on a single moving charge, so it's not surprising that it exerts a force on a current-carrying wire, seeing as a current is a set of moving charges.

Using q = I t, this becomes:

But a velocity multiplied by a time is a length L, so this can be written:

The direction of the force is given by the right-hand rule, where your fingers point in the direction of the current. Current is defined to be the direction of flow of positive charges, so your right hand always gives the correct direction.

sinF qvB sinF IvtB

sinF ILB

The right-hand rule

A wire carries current into the page in a magnetic field directed down the page. In which direction is the force?

1. Left

2. Right

3. Up

4. Down

5. Into the page

6. Out of the page

7. The net force is zero

Three wires

Consider three wires carrying identical currents between two points, a and b.

The wires are exposed to a uniform magnetic field.

Wire 1 goes directly from a to b. Wire 2 consists of two straight sections, one parallel to the magnetic field and one perpendicular to the field. Wire 3 takes a meandering path from a to b. Which wire experiences more force?

1. Wire 1

2. Wire 2

3. Wire 3

4. equal for all three

Three wiresThe force is equal for all three. What matters is the displacement perpendicular to the field, and that's equal for all wires carrying equal currents between the same two points in a uniform magnetic field.

The force on a current-carrying loop

A wire loop carries a clockwise current in a uniform magnetic field directed into the page. In what direction is the net force on the loop?

1. Left

2. Right

3. Up

4. Down

5. Into the page

6. Out of the page

7. The net force is zero

The force on a current-carrying loop

The net force is always zero on a current-carrying loop in a

UNIFORM magnetic field.

Is there a net anything on the loop?

Let’s change the direction of the uniform magnetic field. Is the net force on the loop still zero? Is there a net anything on the loop?

Is there a net anything on the loop?

Let’s change the direction of the uniform magnetic field. Is the net force on the loop still zero? Is there a net anything on the loop?

The net force is still zero, but there is a net torque that tends to make the loop spin.

The torque on a current loopThe magnetic field is in the plane of the loop and parallel to two sides. If the loop has a width a, a height b, and a current I, then the force on each of the left and right sides is F = IbB. The other sides experience no force because the field is parallel to the current in those sides. Simulation

The torque ( ) about an axis running through the center of the loop is:

sinrF

2 2

a aF F

aF

IabB

The torque on a current loop

ab is the area of the loop, so the torque here is .

This is the maximum possible torque, when the field is in the plane of the loop. When the field is perpendicular to the loop the torque is zero. In general, the torque is given by:

IabB IAB

sinIAB

where is the angle between the area vector, A, (which is perpendicular to the plane of the loop) and the magnetic field, B.

A DC motorA direct current (DC) motor is one application of the torque exerted on a current loop by a magnetic field. The motor converts electrical energy into mechanical energy.

If the current always went the same way around the loop, the torque would be clockwise for half a revolution and counter-clockwise during the other half. To keep the torque (and the rotation) going the same way, a DC motor usually has a "split-ring commutator" that reverses the current every half rotation. Simulation

The force between two wires

A long-straight wire carries current out of the page. A second wire, to the right of the first, carries current into the page. In which direction is the force that the second wire feels because of the first wire?

1. Left

2. Right

3. Up

4. Down

5. Into the page

6. Out of the page

7. The net force is zero

The force between two wiresIn this situation, opposites repel and likes attract!

Parallel currents going the same direction attract.

If they are in opposite directions they repel.

A loop and a wire

A loop with a clockwise current is placed below a long straight wire carrying a current to the right. In which direction is the net force exerted by the wire on the loop?

1. Left

2. Right

3. Up

4. Down

5. Into the page

6. Out of the page

7. The net force is zero

A loop and a wire

The long straight wire creates a non-uniform magnetic field, pictured below.

A loop and a wire

The forces on the left and right sides cancel, but the forces on the top and bottom only partly cancel – the net force is directed up, toward the long straight wire.

Five wiresFour long parallel wires carrying equal currents perpendicular to your page pass through the corners of a square drawn on the page, with one wire passing through each corner. You get to decide whether the current in each wire is directed into the page or out of the page.

We also have a fifth parallel wire, carrying current into the page, that passes through the center of the square. Can you choose current directions for the other four wires so that the fifth wire experiences a net force directed toward the top right corner of the square?

How many ways?

You can choose the direction of the currents at each corner. How many configurations give a net force on the center wire that is directed toward the top-right corner?

1. 1

2. 2

3. 3

4. 4

5. 0 or more than 4

How many ways?First, think about the four forces we need to add to get a net force toward the top right. How many ways can we create this set of four forces?

How many ways?How many ways can we create this set of four forces?

Two. Wires 1 and 3 have to

have the currents shown.

Wires 2 and 4 have to

match, so they either both

attract or both repel.

Currents going the same

way attract; opposite

currents repel.

The field from a solenoidA solenoid is simply a coil of wire with a current going through it. It's basically a bunch of loops stacked up. Inside the coil, the field is very uniform (not to mention essentially identical to the field from a bar magnet).

For a solenoid of length L, current I, and total number of turns N, the magnetic field inside the solenoid is given by:

0NIBL

The field from a solenoid

We can make this simpler by using n = N/L as the number of turns per unit length, to get: .

The magnetic field is almost uniform - the solenoid is the magnetic equivalent of the parallel-plate capacitor. If we put a piece of ferromagnetic material (like iron or steel) inside the solenoid, we can magnify the magnetic field by a large factor (like 1000 or so).

0NIBL

0B nI

Magnetism on the atomic level

Currents in wires produce magnetic fields. What produces the magnetic field from a bar magnet, where there are no wires? Why does that field look like the field of a solenoid?

Consider the Bohr model of the atom, where electrons travel in circular orbits around the nucleus. An electron in a circular orbit looks like a current loop, so it produces a magnetic field. In some materials (ferromagnetic materials) the magnetic moments associated with the atoms align, leading to a large net magnetic field.

Whiteboard