19 magnetic fields -...

8866 H1 Physics – J2/2011 10.Electromagnetism

Electromagnetism Content 1. Force on a current carrying conductor 2. Force on a moving charge 3. Magnetic fields due to currents 4. Force between current carrying conductors. Candidates should be able to: (a) show an appreciation that a force might act on a current carrying conductor placed in

a magnetic field. (b) recall and solve problems using the equation F = BIL sin θ, with directions as

interpreted by Fleming’s left hand rule. (c) define magnetic flux density and the tesla (d) show an understanding of how the force on a current carrying conductor can be used

to measure the flux density of a magnetic field using a current balance. (e) predict the direction of a force on a charge moving in a magnetic field (f) sketch flux patterns due to a long straight wire, a flat circular coil and a long solenoid. (g) show an understanding that the field due to a solenoid may be influenced by the

presence of a ferrous core. (h) explain the forces between current carrying conductors and predict the direction of the

forces. References 1. Robert Hutchings, “Physics 2nd Ed”, Nelson, 2000 2. Serway/Faughn, “College Physics 6th Ed,”, Thomson, 2003 3. Loo KW, “ Longman Advanced Level Physics”, Pearson Longman, 2007 4. 2010 J2 H2 Wave Motion Notes by Chua See How ©JJ

tsh_2011

Jurong Junior College /2011 H2 Physics 9646 Electromagnetism

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MAGNETIC FIELD

A field of force is a region of space where there is a force acting on an object placed in that space. An object placed in an ordinary space (like in deep outer space, which is not a field) would not have any force acting on it.

The concept of a magnetic field is the same as the concept of a gravitational field or an electric field. They are special spaces where an object placed in them would have a force acting on that object. Of course, the object must have the appropriate property in order that forces would act on it.

In a gravitational field, the object needs to have mass.

In an electric field, the object needs to have charge.

In a magnetic field, the object needs to have magnetic properties (i.e. like iron and steel) or it is a current carrying conductor or a moving charge.

Fields are produced by objects. A gravitational field is produced by a mass; an electric field is produced by a charge. Note:

a. A magnetic field is produced by a permanent magnet or a current-carrying conductor.

b. A magnetic field can be represented by a diagram of field lines, just like a gravitational or an electric field.

c. Magnetic field is directed from a North pole to a South pole.

A magnetic field is a region of space where a magnetic material, or a current carrying conductor or a moving charge experiences a force when placed in it.

Field lines of a bar magnet Pattern of the magnetic field produced by a bar magnet using iron filings


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A uniform field, within which the field strength is the same at all points, could be represented as parallel lines that are equally spaced as shown above. The uniform field is stronger if the lines are closer to each other as shown below.

Strong field Weak field

The direction of a magnetic field at a point in space is along a tangent to the magnetic field line at that point.

ELECTROMAGNETISM

(a) Show an appreciation that a force might act on a current-carrying conductor placed in a magnetic field.

As mentioned earlier, a gravitational force acts on a mass in a gravitational field; an electric force acts on a charge in an electric field, and a magnetic force might act on a current-carrying conductor in a magnetic field.

Why might, and not definitely would act, like the other two fields? Well, the mass in the gravitational field is a point mass and the charge in the electric field is a point charge. By this, we mean the objects (mass and charge) have no size (no volume), just a point.

For a point mass or charge, there is no direction. The direction of force on the object is determined by the direction of the field.

In the case of a current-carrying conductor, it has a length and the current must flow in a certain direction. Depending on the direction of the current, as compared to the direction of the magnetic field concerned, there is no force if these directions are parallel.

B

Uniform field strength


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(c) Define magnetic flux density and the tesla.

Gravitational Electric Magnetic The field strength field strength flux density

is given by Fgm

= FEq

= ( )

FBsomething

=

Force on an object is given by

F mg= F qE = ( )F something B=

Generally, field strength at a point is expressed as

Field strength = Fo . rce(something)

This (something) in a gravitational field is mass; in an electric field, (something) is

charge. In a magnetic field, (something) is ‘current-carrying conductor’.

• Gravitational field strength at a point, g, is expressed as g = F . m

• Consequently, the gravitational force F acting on an object of mass m placed at that point with gravitational field strength g can be expressed as F = mg.

• Electric field strength at a point, E, is expressed as E = Fq

.

• Consequently, the electric force acting on an object of charge q placed at that point with electric field strength E can be expressed as F =qE.

The term ‘magnetic field strength’ is not used nowadays. The term used to represent field strength in a magnetic field is called ‘magnetic flux density’.

The magnetic flux density at a point, B, is expressed (similar to g and E), as

B = (something)

F .

In symbols, (something) here is expressed as (IL sin θ). Thus, magnetic flux density, B, is expressed as

where, B = magnetic flux density, F = magnetic force acting on the conductor I = current, L = length of conductor, θ = angle between B and I

The unit of B is tesla (symbol: T, named after Nikolai Tesla).

B = sinF

IL θ

Magnetic flux density in a magnetic field is defined as the force per unit length acting on a conductor carrying a unit current placed at right angles to the field.


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i.e. 1 T = o

1 N(1A)(1m)sin 90

(b) Recall and solve problems using the equation F = BlLsinθ, with directions as

interpreted by Fleming’s left-hand rule.

From the equation above, the magnetic force acting on a current-carrying conductor

Hence, Fmax = BIL when

placed at an angle θ to the magnetic field can be expressed as

θ = 900 (I is perpendicular to B) F = 0 when θ = 0 (I is parallel to B)

In Fleming’s left-hand rule, the 3 vectors are represented as such:

ts B, and

Note

• the thumb represents F, • the index finger represen• the middle finger represents I. (in the order of FBI). :

A) F is always perpendicular to B and I,

B) B and I are separated by any angle θ.

F = BILsin θ

Definition of tesla:

he magnetic flux density in a magnetic field is one tesla if one newton of force Tis acting on one metre of a conductor carrying one ampere placed at right angles to the field.


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Example 1: (Ans: D) A horizontal power cable carries a steady current in an east-to-west direction, i.e. into the plane of the diagram. Which arrow shows the direction of the force on the cable caused by the Earth’s magnetic field, in a region where this field is at 70o to the horizontal?

(e) Predict the direction of the force acting on a charge moving in a magnetic field.

The direction of the force acting on a moving charge also follows Fleming’s left-hand rule. Using Fleming’s left-hand rule, the direction of I is equivalent to the flow of positive charges with velocity v in that direction. Thus, I is replaced by the vector v, which represents the velocity of positive charges.

For a negative charge with velocity v, the direction of the force is reversed.

Example 2: (Ans: D) Hot air from a hair – dryer contains many positively charged ions. The motion of these ions constitutes an electric current.

Q

B

v

θ

F

Note: (1) F is always perpendicular to B and v, (2) B and v are separated by any angle θ. (3) For a current-carrying conductor, we have,

sinF BIL θ= , the force F acting on a charge Q moving with a velocity v at an angle θ to the magnetic field of flux density B is given by the equation

F = BQv sin θ


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The hot air is directed between the poles of a strong magnet, as shown. The ions are deflected

A) Towards the north pole N B) Towards the south pole S C) Downwards D) Upwards

(d) Show an understanding of how the force on a current-carrying conductor can be

used to measure the flux density of a magnetic field using a current balance.

A current balance is similar to a beam balance. They both use some ‘weights’ for measurement. In a beam balance, ‘weights’ on one side is balanced by some other ‘weights’ on the other side. In a current balance as shown in E.g. 4, it is balanced by a force on a current-carrying conductor. Current balances come in different designs.

Example 3:

A small square coil of N turns has sides of length L and is mounted so that it can pivot freely about a horizontal axis PQ, parallel to one pair of sides of the coil, through its centre as shown above. The coil is situated between the poles of a magnet which produces a uniform magnetic field of flux density B. The coil is maintained in a vertical plane by moving a rider of mass M along a horizontal beam attached to the coil. When a current I flows through the coil, equilibrium is restored by placing the rider a distance x along the beam from the coil. Starting from the definition of magnetic flux density, show that B is given by

the expression 2

MgxBIL N

= .

Solution: The circuit can be redrawn in side view as follows:

3D view side view

F

F

B

x

coil rider

Mg Pivot PQ


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The flux density B produced by the pair of magnetic poles is directed vertically downwards from N to S. To show how this current balance can be used to measure this B, let us consider taking moments about the pivot PQ: Referring to side view diagram, Clockwise moments due to the weight of the rider = Mg.x Anti-clockwise moments must be due to the forces acting on the square coil. The vertical sides of the coil are parallel to B, so there is no force acting on these vertical sides. For the horizontal sides of the coil, the force F acting on the top side must be to the left and the force acting on the bottom side must be to the right (as indicated on the diagram), such that there would be anti-clockwise moments for balance. The length of each side is L, and each side has N conductors, so the force may be expressed as F = BIL.N. Hence the anti-clockwise moments due to these forces = BILN.L = BIL2N. Equating the moments for balance: BIL2N = Mgx

Transposing, B =NIL

Mgx2

Note: The above example serves to show how one particular design of a current balance may be used to measure B, which in turn depends on the magnetic force acting on current-carrying conductor(s).

(f) Sketch flux pattern due to a long straight wire, a flat circular coil and a long

solenoid.

Right-hand grip rule:

Thumbs : straight direction Fingers : curved direction

Flux Pattern: Flux pattern are diagrams formed by field lines. The 3 flux patterns due to the 3 shapes of conductor (hence current) are as follows:

1. Flux pattern of a magnetic field due to current in a long straight wire:


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2. Flux pattern of a magnetic field due to current in a flat circular coil:

3. Flux pattern of a magnetic field due to current in a long solenoid:

Diagram shows the flux pattern formed by iron filings on a flat horizontal platform due to current in a vertical straight wire through the platform.

Diagram shows a vertical straight wire carrying an upwards current, the directions of the circular magnetic field lines follows the right-hand grip rule.

Diagram shows the flux pattern formed by iron filings on a flat horizontal platform due to current in a flat circular coil in a vertical plane through the platform, its bottom half circle is hidden under the platform.

Diagram shows the magnetic field lines, whose direction follows the right-hand grip rule. The dash lines show the corresponding positions between these diagrams.

Diagram shows the flux pattern formed by iron filings on a flat horizontal platform due to current in a long solenoid; here only the top turns of wire are shown, the bottom turns are hidden under the platform.

Diagram shows the magnetic field lines, whose direction follows the right-hand grip rule.


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In summary,

Shape of conductor

Current direction (I)

Field direction (B) Remarks on field direction

(1) Straight wire

Straight (thumb) Curved (fingers) Changes, depends on position

(2) Flat circular coil

Curved (fingers) Straight (thumb) Straight only along axis of coil

(3) Long solenoid Curved (fingers) Straight (thumb) Straight within solenoid

(g) Show an understanding that the field due to a solenoid may be influenced by the

presence of a ferrous core.

Ferrous core means the material filling the interior of the solenoid is iron.

The magnetic field due to a solenoid would be stronger when it has a ferrous core, i.e. the magnetic flux density in the region near that solenoid would be increased, due to the presence of the ferrous core.

gravitational electric magnetic

The field strength field strength flux density due to a point mass M point charge Q (1) long straight wire

(2) flat circular coil (3) long solenoid

carrying a current I

at a point distance r from M distance r from Q (1) distance d from I (2) distance r from I,

i.e. at centre of coil

(3) within solenoid is expressed as

2

GMg r

= 24 o

QE r

=πε

(1) 2

oId

μ=

πB

(2) 2oNIr

μ=B

(3) oo

NInI

μ= = μB

Remarks:

The 3 equations for B need not be recalled or derived (will be provided if required in question). N is the number of turns in coil or solenoid. n is number of turns per unit length of the long solenoid. μo is the permeability of vacuum. If there is a ferrous core, its permeability μ would be larger than that of vacuum, hence flux density is increased.


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Example 4: (Ans: A) A plotting compass is placed near a solenoid. When there is no current in the solenoid, the compass needle points due north as shown.

When there is a current from X to Y, the magnetic field of the solenoid at the compass is equal in magnitude to the Earth’s magnetic field at that point. In which direction does the plotting compass set?

A

C

B

D

Example 5: (Ans: B) The diagram shows a flat surface with lines OX and OY at right angles to each other.

Which current in a straight conductor will produce a magnetic field at O in the direction OX? A At P into the plane of the diagram B At P out of the plane of the diagram C At Q into the plane of the diagram D At Q out of the plane of the diagram

(h) Explain the forces between current-carrying conductors and predict the direction

of the forces.

In a gravitational field, forces between 2 point masses are always attractive. In an electric field, forces between 2 point charges are as follows: like charges repel, unlike charges attract. In a magnetic field, we consider forces between 2 current-carrying conductors which are parallel and very thin:

• If the currents are in the same direction, they attract, • If the currents are in opposite directions, they repel.


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Comparison between the 3 fields: Field gravitational electric magnetic

name for strength field strength field strength flux density

due to a point mass M point charge Q long straight wire carrying a current I

is expressed as 2

GMg r

= 24 o

QE r

=πε

2

oIB d

μ=

π

force on a point mass m point charge q conductor with

current I, of length at angle θ to

direction of field is expressed as F = mg = qE = BI sin θ

force between 2 masses m1, m2 charges Q1, Q2 currents I1, I2

separated by r r d

is expressed as F

1 22

Gm m

r= 1 2

24 o

QQ r

=πε

1 2

2oI I

d

μ=

π

Note for the last row:

• Forces on very long conductors are very large. • It is more sensible to consider the force on a unit length of a conductor.

• Force per unit length on a conductor, 1 2

2oI IF

dμ

=π

.

• θ = 90o ⇒ sin θ = 1, so equation above does not have sin θ (explained later).

Force acting on masses, charges and current-carrying conductors:

For a gravitational force on a mass m2 in a field due to m1,

1 12 2 2 2( )Gm Gm mF m (g) m

r r= = = 2

For an electric force on a charge Q2 in a field due to Q1,

1 12 2 2 2( )

4 4o o

Q Q QF Q (E) Q r r

= = =πε πε

2

For a magnetic force on a conductor with current I2 in a field due to another parallel conductor with current I1, the force per unit length,

1 12 2( )

2 2o oI IF I (B) I d d

2Iμ μ= = = =

π π

Here, the direction between the B due to I1 and the current I2 are perpendicular, so sin

θ = 1.


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To show the direction of the force, when the 2 currents are in the same direction:

• In the region on the right of I1, the direction of B1 due to I1 is into the paper (represented by x x).

• In this region B1, the direction of the force F acting on I2, by Fleming’s left-hand rule, is to the left, i.e. towards I1. Hence, the force between 2 parallel conductors with currents in the same direction is attractive.

If the direction of I2 is reversed, the direction of the force F acting on it would also be reversed (to the right), i.e. repelled by I1.

If the direction of I1 is reversed, the direction of the field B1 due to I1 would also be reversed (out of paper, represented by • •), so the force F acting on I2 in this reversed field would also be reversed (to the right), i.e. repelled by I1. Hence, the force between 2 parallel conductors with currents in opposite directions is repulsive.

I1 I2 F

B1

I1 I2 F

B1

I1 I2 F

B1


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Example 6: (Ans: C) Two long, parallel wires X and Y carry currents of 3 A and 5 A respectively. The force per unit length experienced by X is 5 x 10-5 N to the right as shown in the diagram (Fig. 18)

The force per unit length experienced by wire Y is A 2 x 10-5 N to the left B 3 x 10-5 N to the right C 5 x 10-5 N to the left D 5 x 10-5 N to the right

Example 7: (Ans: C) Two long, straight, parallel wires carry currents of 1.0 A and 2.0 A. Which diagram shows the directions and relative magnitudes F1 and F2 of the forces per unit length on each of the wires? A

C

B

D


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Some interesting applications of magnetism and electromagnetism

Magnetic Healthcare

Magnets have always been used to promote health. Cleopatra wore jewellery charged with magnetism and Queen Elizabeth 1 is said to have used magnets to ease her arthritis.

And it hasn't gone out of fashion. People who practice magnetic theraphy claim that is natural and simple way of making you feel better is quicker. They say it works on colds and flu, stress, arthritis and rheumatism, headaches, muscle strain, period pain and several chronic skin diseases.

Research has shown that magnetism can make capillary walls relax and blood vessels widen, allowing more blood to pass through. This means the body can get rid of toxins faster and so speed up the healing process.

Electronic Devices

Motors and generators are devices that use electromagnetic induction. Electromagnetic fields are created by electromagnetic induction which states that moving electrical current creates an electronic field. Motors and appliances use electric current in order to create an electromagnetic field to operate the device.

Relays

Relays are how devices such as telephones and computers work. Relays use electromagnets to control the logic and memory function of these devices.

Generators

Electromagnetic fields that move create electrical current. Generators operate off of this principle in order to function and create electrical power for homes. Outside power sources must be used to power the generator in order for the device to create energy.

Transportation

Some trains use electromagnets to elevate the train cars in order for the modern locomotive to move at incredible rapid speeds.

End of lecture notes

19 magnetic fields -...

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