Background knowledge Magnetic field

1
Background knowledge Magnetic field

• Magnetic field can be produced by a magnet or a
  current-carrying conductor.

2
Magnetic field lines

• Magnetic field lines are used to show the
  strength and direction of a magnetic field.

Note 1. Field lines run from the N-pole round to
the S-pole. 2. When field lines are
closely-spaced, field is strong and vice-versa.
3
Example 1 (a)

• In each of the following cases, draw the magnetic
  field lines between the magnets and mark the
  positions of the neutral points (if any).

4
Example 1 (b)

• In each of the following cases, draw the magnetic
  field lines between the magnets and mark the
  positions of the neutral points (if any).

5
Example 1 (c)

• In each of the following cases, draw the magnetic
  field lines between the magnets and mark the
  positions of the neutral points (if any).

6
Example 1 (d)

• In each of the following cases, draw the magnetic
  field lines between the magnets and mark the
  positions of the neutral points (if any).

7
Earths magnetic field

• The Earth has a weak magnetic field. It is so
  weak that it does not affect much the field
  patterns of permanent magnet.
• It is rather like the field around a huge bar
  magnet.
• The south pole of the earth magnet is actually
  in the north.

8
Earths magnetic field

• It is convenient to resolve the earths field
  strength B into horizontal and vertical
  component, BH and BV respectively.

We have BH Bcos a and BV Bsin a. Compass
needles, whose motion is confined to a horizontal
plane, are affected by BH only.
9
Flemings left-hand rule

• F force
• B magnetic field
• I current

10
Factors affecting magnetic force

• B magnetic field
• I current
• l length of conductor in B-field

11
Current balance

• magnetic force F acting upwards
• weight of the rider is equal to the magnetic
  force when the frame PQRS is balanced.

12
magnetic force and current

• Record the currents required to balance different
  number of identical riders on PQ.
• Results magnetic force (number of riders) ?
  current
• i.e. F ? I

13
magnetic force and length

• Keep the current through PQ constant.
• Record the number of riders required to balance
  the metal frame when one, two and three pairs of
  equally strong magnadur magnets are used as
  shown.
• Results
• The magnetic force (number of riders) ? length
  of conductor (number of pairs of magnadur
  magnets)
• i.e. F ? l

14
magnetic force and magnetic field

• Replace the magnadur magnets by a coil of many
  turns (for example 1100 turns) as shown below.
• Hence, the magnetic field is provided from the
  coil and the magnetic field can be increased by
  increasing the coil current.
• Keep the current through PQ constant.
• Record the coil currents required to balance
  different number of identical riders on PQ.

• Results
• The magnetic force (number of riders) ? magnetic
  field (coil current)
• i.e. F ? B

15
Conclusion

• The magnetic force is increased if
• the current is increased, (F ? I)
• the magnetic field is increased, (F ? B)
• there is a greater length of wire inside the
  magnetic field. (F ? l)

16
Precautions of using current balance

• Make sure the direction of the magnetic field is
  perpendicular to the current-carrying arm.
• Minimize the effect of the earths magnetic field
  by aligning the current-carrying arm along the
  N-S direction.
• The set-up should be far from any
  current-carrying conductors so as to avoid the
  effect of stray magnetic fields.
• To avoid overheating, the current should be
  switched off as soon as measurements have been
  taken.
• Shield the set-up from the disturbance of wind.

17
Magnetic flux density B

• Definition
• Electric field E force per unit charge
• (E F / Q)
• Gravitational field g force per unit mass.
• (g F / m)
• Magnetic flux density B (magnetic field) force
  acting per unit current length.
• (B F / Il )
• In words The magnetic flux density B is equal to
  the force acting on a conductor of unit length
  and carrying a unit current at right angles to
  the field.
• Unit of B Tesla (T) or N A-1 m-1

18
Typical Values of the magnetic flux density
Source Magnetic field / T
Smallest value in a magnetically shielded room 10-14
Interstellar space 10-10
Earth's magnetic field 5 x 10-5
Small bar magnet 0.01
Big electromagnet 1.5
Strong lab magnet 10
Surface of a nucleus 106
19
Magnetic force on a current-carrying conductor

• From definition B F/Il
• F BIl
• If the conductor and field are not at right
  angles, but make an angle q with one another, the
  expression becomes
• F BIl sin q.
• Note that
• 1. when q 90o (conductor ? field), F BIl.
• 2. when q 0o (conductor // field), F 0.

20
Magnetic force on moving charge in a magnetic
field

• Current Q / t

21
Magnetic force on moving charge in a magnetic
field

• Magnetic force acting on each moving charge
• If the conductor makes an angle q with the
  magnetic field,
• F Bqv sin q

22
Motion of a charged particle in a magnetic field

• 1 Moving parallel to the field
• When a charged particle moves parallel to the
  magnetic field, i.e. q 0o, there is no force
  acting on it. Its velocity remains unchanged and
  its path is a straight line.

23
2 Moving perpendicular to the field

• Radius of circular path

24
2 Moving perpendicular to the field

• Period of circular motion

25
2 Moving perpendicular to the field

• Notes
• The magnetic force is always perpendicular to the
  motion of the charged particle no work is done
  by the magnetic field.
• Kinetic energy of the charged particle remains
  constant.

26
Example 2

• The path of an electron in a uniform magnetic
  field of flux density 0.01 T in a vacuum is a
  circle of radius 0.05 m. Given that the charge
  and mass of an electron are -1.6 x 10-19 C and
  9.1 x 10-31 kg respectively. Find
• (a) the speed and
• (b) the period of its orbit.
• Solution

27
Current balance

• To measure steady magnetic field by using
  principle of moment.
• Clockwise moment anti-clockwise moment
• mgd2 BIld1

28
Example 3 Current balanceA rider of mass 0.084 g
is required to balance the frame when an arm PQ,
of length 25 cm and carrying a current of 1.2 A,
is inside and in series with a flat, wide
solenoid as shown below.Find the magnetic flux
density inside the solenoid.
29
Search coil and CRO

• A search coil is only used in measuring a varying
  magnetic field.
• A typical search coil consists of 5000 turns of
  wire and an external diameter d 1.5 cm so that
  it samples the field over a small area.
• The search coil is placed with its plane
  perpendicular to a varying magnetic field.

30
Search coil and CRO

• Due to the change of magnetic flux, an e.m.f.,
  which is induced in the search coil, can be
  measured by a CRO.
• The induced e.m.f. E is proportional to the flux
  density B and the frequency f of the varying
  magnetic field. i.e. E ? Bf
• The earths magnetic field can be ignored because
  it is a steady field.

31
Cathode-ray oscilloscope (CRO)

• The CRO is a perfect voltmeter as its resistance
  is very high. It can measure both d.c. and a.c.
  voltages and show how they vary with time.

simulation
32
Measuring d.c. voltages

• To measure a d.c. voltage, the time base is
  usually switched off. Thus, it is the light spot
  which is deflected.
• Or the time base may be switched on to any high
  value so that the horizontal trace is deflected.
  From the deflection on the screen and the gain
  control setting, the d.c. voltage is then
  calculated.

Y-gain control 2 V cm-1
Vd.c. amount of deflection (cm) x Y-gain
control setting (V cm-1) 3 x 2 6 V
33
Measuring a.c. voltages

• To measure an a.c. voltage, the time base is
  usually switched off. The waveform displayed
  becomes a vertical trace so that the amplitude
  can be easily read on the screen. The maximum
  voltage or peak voltage is then calculated from
  the gain control setting. Note that the peak
  voltage refers to half the length of the
  vertical trace.

Y-gain control 5 V cm-1 Time base setting 10 ms
cm-1
Peak voltage Period Frequency
34
Example 4 CRO waveformThe figure below shows a
waveform on a screen.(a) If the controls on the
CRO are set at 0.5 V cm-1 and 2 ms cm-1,
(i) the peak voltage, and (ii) the frequency
of the input signal.Solution
35
Example 4 CRO waveformThe figure below shows a
waveform on a screen.(b) If the gain control is
changed to 1 V cm-1, sketch the trace on the
figure.Solution
36
Example 4 CRO waveformThe figure below shows a
waveform on a screen. (c) If the time base
control is changed to 5 ms cm-1, sketch the
trace on the figure.Solution
37
Time base

• When Vx increases linearly with time from A to B,
  the spot of light on the screen moves at a
  constant speed from the left to right of the
  screen.
• Then the spot of light flies back to the left
  quickly when Vx suddenly drops from B to C.
• The saw-tooth voltage Vx causes no vertical
  movement of the spot of light..

38
Example 5 displaying a.c. waveforms
39
Example 5 displaying a.c. waveforms
40
Measuring of phase relationships

• The phase difference f of two p.ds (Vx and Vy)
  can be observed on phase difference f
• a double-beam CRO.

41

• If a double-beam CRO is not available, the phase
  difference can be found by applying the two p.ds
  of the same frequency and amplitude to the X- and
  Y-plates (time base off) simultaneously.
• The phase difference can be determined from the
  trace on the screen of the CRO as follows.
• In general, the trace is an ellipse except when
  the f is 0o, 90o, 180o, 270o or 360o.

42
Comparing of frequencies

• When two p.ds of different frequencies fx and fy
  are applied to the X- and Y-plates (time base
  off), more complex figures are obtained, know as
  Lissajous figures.

simulation
43
In any particular case, the frequency ratio can
be found by
44
In any particular case, the frequency ratio can
be found by
45
Magnetic field around a long straight wire

• 1. The field lines are circles around the wire.
• 2. The magnetic field is the strongest close to
  the wire.
• 3. Increasing the current makes the magnetic
  field stronger.
• 4. Reversing the current also reverses the
  direction of field lines, but the field pattern
  remains unchanged.

46
Right hand grip rule
If the right hand grips the wire so that the
thumb points the same way as the current, the
fingers curls the same way as the field lines.
47
Experiment to show that B ? I and B ? 1 / r
48

• B ? I and B ? 1 / r
• B ? I/r
• B m0I/(2pr) where I is the current and is the
  permeability of free space (m0 4p x 10-7 T m
  A-1)

49
B m0I/(2pr) (m0 4p x 10-7 T m A-1)

• Example 7
• Two long wires X and Y each carries a current of
  20 A in the directions as shown in the figure. If
  the distance between the wires is 10 mm, find the
  magnitude and direction of the magnetic flux
  density at
• (a) P

50
B m0I/(2pr) (m0 4p x 10-7 T m A-1)

• Example 7
• Two long wires X and Y each carries a current of
  20 A in the directions as shown in the figure. If
  the distance between the wires is 10 mm, find the
  magnitude and direction of the magnetic flux
  density at
• (b) Q.

51
Magnetic field around a flat coil

• At the centre of the coil
• The field lines are straight and at right angles
  to the plane of the coil
• Outside the coil,
• The field lines run in loops.

52
Experimental set-up

• By using the experimental-setup shown, it is
  found that the magnetic field is
• 1. directly proportional to the current and
  the number of turns, and
• 2. inversely proportional to the radius of the
  coil.
• Note The magnetic field is greatest at the
  centre.

53
Magnetic field due to a long solenoid

• From the field pattern of the solenoid, it can be
  found that
• 1. inside the solenoid, the field lines are
  straight and evenly-spaced. This indicates that
  the field is of uniform strength.
• 2. outside the solenoid, the pattern is similar
  to that around a bar magnet, with one end of the
  solenoid behaving like a N-pole and the other
  end like a S-pole.

54
Right hand grip rule
If the right hand grips the solenoid so that the
fingers curls the same way as the current, the
thumbs points to the north pole of the solenoid.
55
Magnetic field due to a long solenoid

• The magnetic field of the solenoid can be
  increased by
• 1. increasing the current,
• 2. increasing the number of turns on the coil.

56
Magnetic field due to a long solenoid

• The magnetic field of the solenoid can be
  increased by
• 1. increasing the current,
• 2. increasing the number of turns on the coil.
• In vacuum, the magnitude of the magnetic flux
  density B at a point O on the axis near the
  centre of the solenoid of length l, having N
  turns and carrying a current I is given by
• B m0NI/l or B m0nI where n is the number of
  turns per unit length (n N/l)

57
Experiment to show that B ? N, B ? I and B ? 1 / l
58
Magnetic field due to a long solenoid

• Note The magnetic field within a solenoid is
  independent of the shape and the cross section
  area of the solenoid.

59
Magnetic field due to a long solenoid

• At the ends of solenoid
• The magnetic field at the ends of the solenoid is
  weaker. It is half that in the central region
  within the solenoid.

60
Current-carrying conductor Position of magnetic field Magnetic flux density (B) Symbol
1. Long straight wire Around the wire r perpendicular distance from wire
2. Circular coil At the centre N number of turns r radius of coil
3. Solenoid Inside
3. Solenoid At the ends
N number of turns l length of solenoid
61
Force between currents

• Magnetic field due to current through P
• Magnetic force acting on Q

Applying Flemings left hand rule, force acting
on wire Q is towards wire P.
62

• Similarly, the force acting on wire P is towards
  wire Q. Hence, the two wires attract each other.
• By Newtons third law, the magnetic force acting
  on P magnetic force acting on Q.
• ?

63
Summary Unlike current repel, like current
attract

• Parallel wires with current flowing in the same
  direction, attract each other.
• Parallel wires with current flowing in the
  opposite direction, repel each other.

64
Summary Unlike current repel, like current
attract

• The force per unit length on each conductor
• When the current I1 I2 1 A, and the
  separation between the wires r 1 m,
• the force per unit length on the conductor

N m-1
65
Summary Unlike current repel, like current
attract

• Definition of the ampere
• The ampere is constant current which, flowing in
  two infinitely long, straight, parallel
  conductors of negligible cross-section and placed
  in a vacuum 1 metre apart, produces between them
  a force equal to 2 x 10-7 Newton per metre of
  their length.

66
Example 8

• The figure below shows two horizontal wires, P
  and Q, 0.2 m apart, carrying currents of 1.5 A
  and 3 A respectively.
• (a) Calculate the force per metre on wire Q.
• (b) State the direction of the force.
• (c) State the direction and magnitude of the
  force per metre on wire P due to the current in
  wire Q

.
67
Moment and couple

• Couple - consists of 2 equal and opposite
  parallel forces whose lines of action do not
  coincide (??).

F
d
F
torque of couple F x d/2 F x d/2 Fd
68
Torque on a rectangular current-carrying coil in
a uniform B-field

• Torque F(b sin q)
• NBIlb sin q
• NBAI sin q

69
Maximum and minimum torque on a coil

• The maximum torque is NBAI when the plane of coil
  is parallel to the field (q 90o).
• The torque on the coil is zero when the plane of
  coil is perpendicular to the field (q 0o).

70
Example 9

• Example 9
• A square coil has sides of length 5 cm. The coil
  consist of 20 turns of insulated wire carrying a
  current of 0.2 A. The plane of the coil is at an
  angle 40o to a uniform magnetic field of flux
  density of 25 mT. Calculate the torque acting on
  the coil.
• Solution

71
Moving-coil galvanometer

• The moving-coil meter contains a coil wound on an
  aluminium former around a soft-iron cylinder.
• The coil is pivoted on bearings between the poles
  of a cylindrical magnet.
• Current flows through the coil via a pair of
  spiral springs called hair springs.

72
Theory

• When a current is passed through a coil in a
  magnetic field, the coil experiences a torque.
  The coil rotates, moving the pointer across the
  scale.
• The normal of plane of the coil is always
  perpendicular to the magnetic field, the torque
  on the coil is given by
• T NBAI sin 90o NBAI

73
Theory

• The movement of the coil is opposed by the hair
  springs.
• The restoring torque (t) exerted by the hair
  springs to oppose the rotation is given by
• t kq
• where k is the torsion constant of the
  hairsprings

74
Linear scale

• The coil comes to rest when the magnetic turning
  effect (torque) on the coil is balanced by the
  turning effect (restoring torque) from the hair
  springs.
• BANI kq
• Hence, I ? q the galvanometer scale is linear

75
Sensitivity

• The current sensitivity of a galvanometer is
  defined as the deflection per unit current
• i.e. current sensitivity q/I BAN/k.
• The voltage sensitivity of a galvanometer is
  defined as the deflection per unit voltage
• i.e. voltage sensitivity q/V q/(IR)
  BAN/(kR)
• where R is the resistance of the coil.
• The sensitivity can be increased, i.e. the coil
  deflects more for a certain current or voltage,
  by
• 1. using a stronger magnet (larger B),
• 2. using weaker hair springs (smaller k).
• 3. using a coil with larger area (larger A),
  and
• 4. increasing the number of turns of the coil
  (larger N).