Fields and forcesFields and forces

Topic 6.3: Topic 6.3: Magnetic force and fieldMagnetic force and field

The magnetic field around a long The magnetic field around a long straight wirestraight wire

The diagram shows a wire carrying a The diagram shows a wire carrying a current of about 5 ampscurrent of about 5 amps

If you sprinkle some iron filings on to the If you sprinkle some iron filings on to the horizontal card and tap it gently, the iron horizontal card and tap it gently, the iron filings will line up along the lines of flux as filings will line up along the lines of flux as shown. shown.

You can place a small compass on the You can place a small compass on the card to find the direction of the magnetic card to find the direction of the magnetic field.field.

With the current flowing up the wire, the With the current flowing up the wire, the compass will point anti‑clockwise, as compass will point anti‑clockwise, as shown.shown.

What will happen if you reverse the What will happen if you reverse the direction of the current?direction of the current?

The diagrams show the magnetic field The diagrams show the magnetic field as you look down on the cardas you look down on the card

Imagine the current direction as an Imagine the current direction as an arrow.arrow.

When the arrow moves away from you, When the arrow moves away from you, into the page, you see the cross (x) of into the page, you see the cross (x) of the tail of the arrow.the tail of the arrow.

As the current flows towards you, you As the current flows towards you, you see the point of the arrow ‑ the dot in see the point of the arrow ‑ the dot in the diagram.the diagram.

Can you see that the further from the Can you see that the further from the wire the circles are, the more widely wire the circles are, the more widely separated they become? What does this separated they become? What does this tell you?tell you?

The flux density is greatest close to the The flux density is greatest close to the wire.wire.

As you move away from the wire the As you move away from the wire the magnetic field becomes weaker.magnetic field becomes weaker.

The 1The 1stst right‑hand right‑hand rule rule gives a simple gives a simple way to remember the way to remember the direction direction of the field: of the field:

imagine gripping the imagine gripping the wire, so that your wire, so that your right right thumb thumb points in the points in the direction of the current.direction of the current.

your fingers then curl your fingers then curl in the direction of the in the direction of the lines of lines of the fieldthe field::

The magnetic field of a flat The magnetic field of a flat coilcoil

The diagram shows a flat coil carrying The diagram shows a flat coil carrying electric current:electric current:

Again, we can investigate the shape and Again, we can investigate the shape and direction of the magnetic field using iron direction of the magnetic field using iron filings and a compass.filings and a compass.

Close to the wire, the lines of flux are circles.Close to the wire, the lines of flux are circles. Can you see that the lines of flux run Can you see that the lines of flux run

anti‑clockwise around the left side of the coil anti‑clockwise around the left side of the coil and clockwise around the right side?and clockwise around the right side?

What happens at the centre of the coil?What happens at the centre of the coil? The fields due to the sides of the coil are in The fields due to the sides of the coil are in

the same direction and they combine to give the same direction and they combine to give a strong magnetic field.a strong magnetic field.

How would you expect the field to change, if How would you expect the field to change, if the direction of the current flow around the the direction of the current flow around the coil was reversed?coil was reversed?

The magnetic field of a The magnetic field of a solenoidsolenoid

A solenoid is a long coil with a large A solenoid is a long coil with a large number of turns of wire.number of turns of wire.

Look at the shape of the field, revealed by Look at the shape of the field, revealed by the iron filings.the iron filings.

Does it look familiar?Does it look familiar?

The magnetic field The magnetic field outside outside the solenoid has the the solenoid has the same shape as the field around a bar same shape as the field around a bar magnet.magnet.

Inside Inside the solenoid the lines of flux are close the solenoid the lines of flux are close together, parallel and equally spaced.together, parallel and equally spaced.

What does this tell you?What does this tell you? For most of the length of the solenoid the flux For most of the length of the solenoid the flux

density is constant.density is constant. The field is uniform and strong.The field is uniform and strong.

If you reverse the direction of the current If you reverse the direction of the current flow, will the direction of the magnetic field flow, will the direction of the magnetic field reverse?reverse?

A right‑hand grip A right‑hand grip rule can again be rule can again be used to remember used to remember the direction of the the direction of the field, field, but but this time this time you must curl the you must curl the fingers of your fingers of your right right hand in the hand in the direction of the direction of the current as shown:current as shown:

Your thumb now points along the direction Your thumb now points along the direction of the lines of flux of the lines of flux inside inside the coil . . . the coil . . . towards the end of the solenoid that towards the end of the solenoid that behaves like the N‑pole of the bar magnet.behaves like the N‑pole of the bar magnet.

This 2This 2ndnd right‑hand rule can be used for the right‑hand rule can be used for the flat coil. Thumb in the direction of the flat coil. Thumb in the direction of the current and fingers curl in the direction of current and fingers curl in the direction of the magnetic fieldthe magnetic field

Magnetic Forces – on WiresMagnetic Forces – on Wires

A wire A wire carrying a carrying a current in a current in a magnetic field magnetic field feels a force.feels a force.

A simple way A simple way to to demonstrate demonstrate this is shown this is shown in the diagramin the diagram

The two strong magnets are attached to The two strong magnets are attached to an iron yoke with opposite poles facing an iron yoke with opposite poles facing each other.each other.

They produce a strong almost uniform field They produce a strong almost uniform field in the space between them.in the space between them.

What happens when you switch the What happens when you switch the current on?current on?

The aluminium rod AB feels a force, and The aluminium rod AB feels a force, and moves along the copper rails as shown.moves along the copper rails as shown.

Notice that the current, the magnetic Notice that the current, the magnetic field, and the force, are all at right field, and the force, are all at right angles to each other.angles to each other.

What happens if you reverse the What happens if you reverse the direction of the current flow, or turn the direction of the current flow, or turn the magnets so that the magnetic field acts magnets so that the magnetic field acts downwards?downwards?

In each case the rod moves in the In each case the rod moves in the opposite direction.opposite direction.

Why does the aluminium rod move?Why does the aluminium rod move?The magnetic field of the permanent The magnetic field of the permanent

magnets interacts with the magnetic field magnets interacts with the magnetic field of the current in the rod.of the current in the rod.

Imagine looking from end B of the rod.Imagine looking from end B of the rod.The diagram shows the combined field of The diagram shows the combined field of

the magnet and the rodthe magnet and the rod

The lines of flux behave a bit like elastic bands.The lines of flux behave a bit like elastic bands. Can you see that the wire tends to be catapulted Can you see that the wire tends to be catapulted

to the left?to the left? You can use You can use the third right‑hand rule the third right‑hand rule to predict to predict

the direction of the force. You need to hold your the direction of the force. You need to hold your right hand so that the thumb points in the right hand so that the thumb points in the direction of the current, the fingers point in the direction of the current, the fingers point in the direction of the magnetic field, then the palm of direction of the magnetic field, then the palm of your hand gives the direction of the magnetic your hand gives the direction of the magnetic forceforce

Calculating the ForceCalculating the Force

Experiments like this show us that the Experiments like this show us that the force F on a conductor in a magnetic field force F on a conductor in a magnetic field is directly proportional to:is directly proportional to:

the magnetic flux density Bthe magnetic flux density B the current I,the current I,and the length and the length LL of the conductor in the of the conductor in the

field.field.

In fact:In fact:

This equation applies when the current is at 90° This equation applies when the current is at 90° to the field.to the field.

Does changing the angle affect the size of the Does changing the angle affect the size of the force?force?

Look at the wire OA in the diagram, at different Look at the wire OA in the diagram, at different angles:angles:

When the angle When the angle θθ is 90° the force has its is 90° the force has its maximum value.maximum value.

As As θθ is reduced the force becomes smaller. is reduced the force becomes smaller. When the wire is parallel to the field, so that When the wire is parallel to the field, so that θθ is is

zero, the force is also zero.zero, the force is also zero. In fact, if the current makes an angle In fact, if the current makes an angle θθ to the to the

magnetic field the force is given by:magnetic field the force is given by:

Notice that: when Notice that: when θθ = 90°, sin = 90°, sin θθ = 1, = 1, and and F = B I F = B I LL as before.as before.when when θθ = 0°, sin = 0°, sin θθ = 0, = 0, and and F = 0, F = 0, as stated above.as stated above.

The size of the force depends on the The size of the force depends on the angle that the wire makes with the angle that the wire makes with the magnetic field, but the magnetic field, but the direction direction of the of the force does not.force does not.

The force is always at 90° to both the The force is always at 90° to both the current and the field.current and the field.

Magnetic flux density B and the Magnetic flux density B and the teslatesla

We can rearrange the equation We can rearrange the equation F = F = B B II LL to give:to give:

B = F /IB = F /ILLWhat is the value of What is the value of B, B, when I = 1 A and when I = 1 A and LL

= 1 m?= 1 m? In this case, In this case, B B has the same numerical has the same numerical

value as value as F.F.

This gives us the definition of This gives us the definition of B:B: The magnetic flux density The magnetic flux density B, B, is the force acting is the force acting

per unit length, on a wire carrying unit current, per unit length, on a wire carrying unit current, which is perpendicular to the magnetic field.which is perpendicular to the magnetic field.

The unit of B is the The unit of B is the tesla tesla (T).(T). Can you see that: 1 T = 1 N ACan you see that: 1 T = 1 N A‑1‑1 m m‑1‑1 ? ? The tesla is defined in the following way:The tesla is defined in the following way: A magnetic flux density of 1 T produces a force A magnetic flux density of 1 T produces a force

of 1 N on each metre of wire carrying a current of 1 N on each metre of wire carrying a current of 1A at 90° to the field.of 1A at 90° to the field.

Magnetic Forces – on ChargesMagnetic Forces – on Charges A charged particle feels a force when it moves A charged particle feels a force when it moves

through a magnetic field.through a magnetic field. What factors do you think affect the size of this What factors do you think affect the size of this

force?force?

The force F on the particle is directly The force F on the particle is directly proportional to:proportional to:

the magnetic flux density B,the magnetic flux density B, the charge on the particle Q, andthe charge on the particle Q, and the velocity v of the particle.the velocity v of the particle.

When the charged particle is moving at 90° to When the charged particle is moving at 90° to the field, the force can be calculated from:the field, the force can be calculated from:

In which direction does the force act?In which direction does the force act? The force is always at 90° to both the current The force is always at 90° to both the current

and the field, and you use and the field, and you use the 3the 3rdrd right‑hand right‑hand rule rule to find its direction.to find its direction.

(Note: the right‑hand rule applies to (Note: the right‑hand rule applies to conventional current flow.) conventional current flow.)

A negative charge moving to the right, has to A negative charge moving to the right, has to be treated as a positive charge moving to the be treated as a positive charge moving to the left. left.

You must point your thumb in the opposite You must point your thumb in the opposite direction to the movement of the negative direction to the movement of the negative charge.charge.

This equation applies when the direction of This equation applies when the direction of the charge motion is at 90° to the field.the charge motion is at 90° to the field.

Does changing the angle affect the size of Does changing the angle affect the size of the force?the force?

As As θθ is reduced the force becomes is reduced the force becomes smaller.smaller.

When the direction is parallel to the field, When the direction is parallel to the field, so that so that θθ is zero, the force is also zero. is zero, the force is also zero.

In fact, if the charge makes an angle In fact, if the charge makes an angle θθ to to the magnetic field the force is given by:the magnetic field the force is given by:

F = QvB sin F = QvB sin θθ

2 Parallel Current-Carrying Wires2 Parallel Current-Carrying Wires

What happens when current is What happens when current is passed along two strips of foil passed along two strips of foil as shown below?as shown below?

The strips bend, as they attract The strips bend, as they attract or repel each other.or repel each other.

Two parallel, current‑carrying Two parallel, current‑carrying wires exert equal, but opposite wires exert equal, but opposite forces on each other.forces on each other.

Look carefully at these forces, Look carefully at these forces, and the resultant magnetic and the resultant magnetic fields around the wires.fields around the wires.

The RulesThe Rules

currents flowing in the currents flowing in the same same direction direction attractattract currents flowing in currents flowing in opposite opposite directions directions repel.repel.

How do these How do these forces arise?forces arise?

The diagram The diagram shows the shows the anti‑clockwise anti‑clockwise field around wire field around wire X:X:

Wire Y is at 90° to this field, and so it Wire Y is at 90° to this field, and so it experiences a force.experiences a force.

Apply 3Apply 3rdrd right‑hand rule to wire Y right‑hand rule to wire YDo you find that the force on wire Y is to Do you find that the force on wire Y is to

the left, as shown?the left, as shown?

What is the size of the force?What is the size of the force? Notice that the wires are a distance r apart. Notice that the wires are a distance r apart. Wire X carries a current IWire X carries a current Ill. Wire Y carries a current . Wire Y carries a current II22..

What is the flux density What is the flux density B B at wire Y, due to the current Iat wire Y, due to the current I ll

in X?in X? From B = From B = μμ00II11/2/2ππrr What is the force F on a length What is the force F on a length LL of wire Y? of wire Y? From F = BIFrom F = BI22LL If we use the first equation to replace BIf we use the first equation to replace B in the second in the second

equationequation

Defining the AmpereDefining the Ampere

The unit of current, the ampere, is defined The unit of current, the ampere, is defined in terms of the force between two currents.in terms of the force between two currents.

When two long wires are parallel, and When two long wires are parallel, and placed 1 metre apart in air, and if the placed 1 metre apart in air, and if the current in each wire is 1 ampere, then the current in each wire is 1 ampere, then the force on each metre of wire is 2 x 10force on each metre of wire is 2 x 10‑7‑7 N. N.

ThereforeTherefore

Thus, one ampere is defined as that Thus, one ampere is defined as that current flowing in each of two current flowing in each of two infinitely‑long parallel wires of negligible infinitely‑long parallel wires of negligible cross‑sectional area separated by a cross‑sectional area separated by a distance of one metre in a vacuum that distance of one metre in a vacuum that results in a force of exactlyresults in a force of exactly 2 x 10 2 x 10‑7‑7 N N per metre of length of each wire.per metre of length of each wire.

The Magnetic Field Due to CurrentsThe Magnetic Field Due to Currents

The The magnetic field intensity or magnetic field intensity or magnetic induction or magnetic flux magnetic induction or magnetic flux density density is given the symbol B and it has is given the symbol B and it has thethe units of units of tesla Ttesla T

It is a vector quantity.It is a vector quantity.

The strength of the magnetic field B at any point The strength of the magnetic field B at any point at a perpendicular distance r from a long straight at a perpendicular distance r from a long straight conductor carrying a current I is given byconductor carrying a current I is given by

where μ0 = 4π x 10 -7 T m A-1 is a constant called the permeability of free space.

For a Long Straight Conductor For a Long Straight Conductor

For a SolenoidFor a Solenoid

If the solenoid has N turns, length If the solenoid has N turns, length LL and and carries a current I, the flux density B at a point carries a current I, the flux density B at a point O on the axis near the centre of the solenoid, O on the axis near the centre of the solenoid, is found to be given byis found to be given by

B = B = μμ00 N NII / / LL

Or B = Or B = μμ00 n nII

where where n = Nn = N // L L = number of turns per unit = number of turns per unit length.length.

B thus equals B thus equals μμ00 , multiplied by the , multiplied by the ampere‑turns per metre.ampere‑turns per metre.

The Nature of the Solenoid The Nature of the Solenoid CoreCore

Be aware that the nature of the solenoid Be aware that the nature of the solenoid core has an affect on Bcore has an affect on B

An iron core concentrates the magnetic An iron core concentrates the magnetic field thus making B greater.field thus making B greater.

If a steel core is used it is not turn-off-ableIf a steel core is used it is not turn-off-ableAn electromagnet is good because it can An electromagnet is good because it can

be turned on and off, and can have its be turned on and off, and can have its strength varied.strength varied.