3'4ELECTROMAGNETISM

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3.4 ELECTROMAGNETISM
3.4.1 Magnetic Force on Currents
3.4.2 Hall Effect
3.4.3 Magnetic Fields due to Currents
3.4.4 Torque on a Coil in a Magnetic Field
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3.4.1 Magnetic Force on Currents

• Moving charges in a magnetic field
• A magnetic field is a region in which a
  charge would experience a magnetic force.
• In the presence of both an electric and a
  magnetic field, a moving charge would experience
  both the and force.
• Magnetic field representation
• A magnetic field B is represented by
  .
• Unlike electric field lines, magnetic field lines
  can form .
• The direction of field is indicated by the
  on the line.
• The number of lines per unit cross-sectional area
  is proportional to the of the field.

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• Simple current balance
• The following figure shows a simple
  to investigate magnetic force on a
  current-carrying wire.
• The size of magnetic force is equal to the
  of the .
• How can the (i) size of magnetic field (ii)
  length of conductor experiencing the field (iii)
  current in the coil and (iv) angle between
  direction of field and current be varied in the
  experiment?

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• Definition of magnetic flux density B
• The magnetic flux density B is defined as the
  force acting per on a conductor carrying
  at to the direction of the magneic
  field. In symbols,
• B
• If F 1 N when I 1 A and ? 1 m, then B 1 N
  A-1m-1, which is given the name .
• Measuring B using a current balance
• The current balance enables us to measure the
  flux density of field produced by various
  current-carrying elements.
• What is the flux density inside a solenoid if a
  rider of mass 0.084 g is needed to balance the
  force on 25 cm wire of current 1.2 A inside the
  solenoid?

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• Expression for force on a current-carrying
  conductor
• The magnetic force F acting on a wire carrying a
  current I with a length ? in a magnetic field B
  is given by
• F Eq.(2)
• where the direction of the current vector I is
  taken the same as that of the .
• The magnitude of F in terms of ? .
• The direction of F is determined by the
  hand grip rule.
• If ? 90o, the magnitude of the magnetic force
  becomes , and the direction can be
  determined by the ( ).

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• Path of a moving charge in a uniform magnetic
  field
• v parallel to B
• If the charge enters the field in a direction
  parallel to the field, the magnetic force on the
  charge would be since ? . The charge would
  travel with speed
• v perpendicular to B
• If the charge enters the field in a direction
  perpendicular to the field, the magnetic force on
  the charge would be in a direction
  to both B and v. The charges path is
  continually by the magnetic force,
  performing motion of radius with a period
  .

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• v at an angle to B
• If the charge enters the field at an angle ? to
  the field, the motion can be studied by resolving
  the motion into one that are parallel and
  perpendicular to the magnetic field. The motion
  of the charge parallel to the field is by
  the magnetic field, while that perpendicular to
  the field is subjected to a constant deflecting
  force, which provides the force for
  motion in the plane perpendicular to the . The
  resultant path of the charge is therefore a ,
  with radius , and pitch length .

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Examination questions

• 1993-IIA-49
• Two particles P and Q of same quantity of charge
  and mass but moving with different speeds vP and
  vQ respectively enter a region of uniform
  magnetic field directed into the plane of the
  paper. The subsequent circular paths are as
  shown. Which of the following statements is/are
  correct?
•   (1) Both P and Q are positively charged.
• (2) vP is smaller than vQ.
• (3) The period of circular motion of P is
  shorter than that of Q.
•   A. (1) only B. (3) only C. (1)
  and (2) only
• D. (2) and (3) only E. (1), (2) and (3)

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• 1995-IIA-34
• Particles A and B moving at the same speed
  enters a square region of uniform magnetic field
  as shown. Particles A leaves at X while particle
  B leaves at Y. If the charge to mass ratio of
  particle A is k, what is that of particle B?
• A. k/2 B. k/4 C. k D. 2k
  E. 4k
• 1998-IIA-26
• The figure shows a charged particle moving in a
  circle
• with constant speed v on a plane
  perpendicular to a
• uniform magnetic field. Which of the following
  graphs
• represents the relation between the time T for
  the particle to complete a circle and its speed
  v?
• A. B. C. D. E.

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• 2002-IIA-21
• The above figure shows an electron entering a
  uniform field which may be electric or magnetic.
  Which of the following descriptions about the
  subsequent motion of the electron is correct?
• A. Only in a magnetic field can the electron be
  deflected by more than 900.
• B. Only in an electric field does the force
  depend on the magnitude of the charge on the
  electron.
• C. Whether the field is electric or magnetic,
  the speed of the electron will increase.
• D. Whether the field is electric or magnetic,
  the magnitude and direction of the force acting
  on the electron are constant.

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• 2003-IIA-31
• A charged particle enters a region of uniform
  magnetic field whose direction is normal to the
  initial velocity of the particle. The subsequent
  path of the particle is as shown.
• Which of the following is/are possible
  explanations to account for the shape of the
  path?
• (1) The flux density of the magnetic field has
  decreased gradually.
• (2) The charged particle has lost its charge
  gradually.
• (3) The charged particle has lost kinetic energy
  gradually.
• A. (1) only
• B. (3) only
• C. (1) and (2) only
• D. (2) and (3) only

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• Charge across a crossed electric and magnetic
  field
• When a beam of charges passes through a crossed
  electric and magnetic field in a diretion
  perpendicular to both fields, each charge is
  subjected to and .
• The charge is undeflected if , i.e., when

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Examination questions

• 2000-IIA-29
• A beam of charged particles passes undeflected
  through a region of crossed uniform electric and
  magnetic fields. Which of the following must be
  common to the particles making up this beam?
• A. charge to mass ratio
• B. velocity
• C. mass
• D. sign of charge
• E. magnitude of charge

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3.4.2 Hall Effect

• Explanation
• In the presence of magnetic field, charge
  carriers experience forces that are to
  both the and the .
• Owing to the deflecting force, one side of the
  conductor thus develops sign that of
  (majority) charge carriers, setting up an
  field in a direction to both the and .
  
• A small is said to be established across the
  conductors sides, in a direction to both
  the and the .

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• Expression for hall voltage
• Magnetic force acting on a charge q moving with
  speed vd in the magnetic field B .
• Electric force acting on a charge q by the
  electric field E due to hall effect .
• Steady state is reached when the force
  has grown to a size the same as the
  force, i.e., when , or .
• But, in terms of hall voltage VH, electric field
  E , while in terms of current I, drift speed
  vd .
• Thus,
• VH

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• Hall probe
• In the above circuit, a current of about 50 mA is
  passed through the Hall slice, a piece of doped
  with charge carriers, via terminals P and
  Q.
• If the hall p.d. connections X and Y on the slice
  are not each other, a p.d. exists between
  them even in the absence of a magnetic field and
  there will be a current through the voltmeter.
• In this case the balance control should be used
  to set the meter and bring X and Y to the
  same .
• To measure the magnetic field, one should
  orientate the probe until a voltage is
  measured.

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Examination questions

• 1995-IIA-32
• In Hall probe, the slice of semiconductor
  inside has 1025 charge-carriers per cubic metre.
  When a steady current of 0.4 A passes through the
  slice and a uniform magnetic field of 0.1 T
  applies perpendicularly to it, a Hall voltage of
  20 ?V is set up. Find the thickness of the
  slice. (Given electronic charge 1.6 ? 10-19 C)
• A. 0.9 ? 10-3 m B. 1.1 ? 10-3 m C. 1.3 ?
  10-3 m
• D. 1.5 ? 10-3 m E. 1.7 ? 10-3 m
• 2004-IIA-26
• The Hall effect
• (1) provides evidence that the charge carriers
  in metals are negatively charged.
• (2) can be used to determine the density of free
  electrons in metals.
• (3) can be used to determine the flux density of
  the magnetic field due to an a.c. only.
•   Which of the above statements are correct?
• A. (1) and (3) only B. (1) and (2) only C.
  (2) and (3) only
• D.(1), (2) and (3)

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3.4.3 Magnetic Fields due to Currents

• The Biot-Savart law
• Theoretically, Biot and Savart stated that for a
  very small charge q moving with velocity v, the
  flux density B at a point P located at r relative
  to the charge is
• B
• where is the permeability of the medium
  considered.
• The permeability of a vacuum is denoted by ,
  and its numerical value is .
• In most cases, the calculation of flux density
  requires the use of in which flux density due
  to differential parts of the conductor are
  integrated.

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• Magnetic fields around a long straight wire
• The magnitude of the flux density at point P
  distant r from a very long straight wire carrying
  a current I in vacuum is
• B
• Magnetic fields inside a long solenoid
• The magnitude of the flux density at point
  inside a very long solenoid having n turns per
  unit length carrying a current I is
• B

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• Magnetic forces between currents
• Magnitude of magnetic field at the right-hand
  conductor due to current I1 in the left-hand one
  is
• B1
• The force F21 acting on length ? of the
  right-hand conductor is
• F21
• The force F12 acting on length ? of the left-hand
  conductor has magnitude but direction as
  F21.

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• Definition of ampere
• The definition of the ampere is based on the
  previous expression and may be stated as follows
• The ampere is the constant current which ,
  flowing in two infinitely , and
  conductors of negligible , placed in a vacuum
  m apart, produces between then a force of
  N m 1 of the conductors.
• Absolute measurement of current
• What is the current flows
• in the wire?

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Examination questions

• 1991-IIA-34
• The magnetic field due to a steady current in
  the long air-core solenoid (uniformly wound)
  shown below is 8 ? 10-3 T at X and 1 ? 10-3 T at
  Y.
• If an identical solenoid is connected to the end
  of the first so as to extend it to as far as Y,
  and the same current as before is passed through
  the complete solenoid, the magnetic field at Y
  will then be
• A. 7 ? 10-3 T
• B. 8 ? 10-3 T
• C. 9 ? 10-3 T
• D. 16 ? 10-3 T
• E. 17 ? 10-3 T

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• 1993-IIA-36
• The above figure shows two long parallel
  straight wires separated by a distance of 0.2 m,
  carrying currents of 1 A in opposite directions.
  The magnetic field at a point X mid-way between
  the wires is (Given permeability constant ?o
  4? ? 10-7 T m A-1)
• A. 0 T B. 2 ? 10-7 T into paper
  C. 2 ? 10-7 T out of paper
• D. 4 ? 10-6 T into paper E. 4 ? 10-6 T out of
  paper
• 1994-IIA-31
• For two long, straight parallel conducting wires
  carrying the same current, the magnitude of the
  force acting on a section of the wires depends on
  
•   (1)     the distance between the wires
• (2)     the length of that section of wires
• (3)     the directions of current flow in the
  wires
•   A. (1) only B. (3) only C. (1) and
  (2) only
• D. (2) and (3) only E. (1), (2) and (3)

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• 1996-IIA-26
• Four infinitely long straight parallel wires P,
  Q, R, S carrying equal currents are situated at
  the corners of a square as shown. The currents
  in P and Q are into paper and those in R, S are
  out of paper. What is the direction of the
  resultant magnetic induction at the centre of the
  square?
•   A. I B. II C. III D. IV E. V
• 1998-IIA-26
• Two long, straight parallel conducting wires,
  each carrying a current I, are separated by a
  distance r as shown. What is the magnetic field
  at a point P at the same distance r from both
  wires?
• (?o permeability of free space)
• A. ?oI/2?r to the left B. ?3?oI/2?r to
  the left C. ?oI/?r to the left
• D. ?3?oI/2?r to the right E. ?oI/2?r to the
  right

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• 1999-IIA-31
• For which of the following does the force
  between two objects vary inversely as the square
  of the distance between their centres?
•   (1) Two equal masses joined by a rubber band
  under tension.
• (2) Two long straight parallel conducting wires
  carrying steady electric currents.
• (3) The earth and a satellite in its orbit.
•   A. (1) only
• B. (3) only
• C. (1) and (2) only
• D. (2) and (3) only
• E. (1), (2) and (3)

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• 1999-IIA-35
• Two long, straight, parallel conducting wires P
  and Q are positioned as shown. The same current
  flows through both wires and is directed into the
  plane of the paper. Points A, B and C on the
  plane of the paper are equidistant from both
  wires where C is the mid-point between the wires.
  Which of the following statements is/are
  correct?
•   (1) The magnetic field strength at C is greater
  than that at A.
• (2) The directions of the magnetic field at A
  and at B are the same.
• (3) The magnetic field strength at B will
  increase if the current flowing in the wires
  increases.
•   A. (1) only B. (3) only C. (1) and (2)
  only
• D. (2) and (3) only E. (1), (2) and (3)

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• 2000-IIA-25
• Four parallel long straight conductors carrying
  currents of equal magnitude pass vertically
  through the four corners of a square PQRS. The
  current is directed into paper in one conductor
  and is directed out of paper in the other three
  conductors. Which of the following arrangement
  can produce a resultant magnetic induction at the
  centre O of the square in the direction shown?
•   Current into paper Current out of paper
• A. P Q, R, S
• B. Q P, R, S
• C. R P, Q, S
• D. S P, Q, R
• E. It is impossible to produce a resultant
  magnetic induction at O in the direction shown.

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• 2002-IIA-29
• Three long straight parallel wires P, Q and R
  carrying
• currents of the same magnitude are situated at
  the
• vertices of an equilateral triangle as shown.
  The
• currents in wires P and R are directed out of
  the
• paper. Which of the following indicates the
  direction of the force acting on wire P?
• A. B. C. D.
• 2003-IIA-30
• In the above set-up, AB and CD are two
• parallel infinitely long wires 20 cm apart,
• carrying currents I1 and I2 respectively. The
• magnitude flux density at the point P 10 cm from
  wire CD is zero. If I2 0.6 A, which of the
  following statements about I1 is correct?
• A. 0.2 A flows in the same direction as I2.
• B. 0.2 A flows in the opposite direction as I2.
• C. 1.8 A flows in the same direction as I2.
• D. 1.8 A flows in the opposite direction as I2.

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• 2004-IIA-29
• Three long, parallel, straight current-carrying
  wires P,Q and R are placed in the same plane in
  air as shown.
• If F is the force per unit length between two
  long, parallel, straight wires placed a distance
  d apart and each carrying a current of 1 A, what
  is the net force per unit length acting on the
  wire R shown?
•  
• A. 0
• B. F
• C. 2F
• D. 3F

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3.4.4 Torque on a Coil in a Magnetic Field

• Expression for magnetic torque
• Consider a rectangular coil of N turns and face
  area A ? ? b carrying current I pivoted so that
  it can rotate about a vertical axis which is at
  right angles to a uniform magnetic field of flux
  density B as shown. Let the normal to the plane
  of the coil makes an angle ? with the field.
• Force on each vertical side of length ? F
• Torque (couple) on the coil ?
• ? Eq.(2)

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• Extension of expression to coil of any shape
• The expression for ? can be shown to hold for a
  coil of shape of area A.
• A coil of arbitary shape carrying current I can
  be regarded as consisting of a large number of
  , each with current flowing in the
  same sense as in the large coil.
• The forces on all sides of the tiny coils
  except where they lie on the large coil itself.

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• Moving-coil galvanometer
• A moving-coil galvanometer consists of a coil of
  copper wire that is able to rotate in a strong
  magnetic field which is produced in the narrow
  gap between the of a permanent magnet
  and a fixed . The field line in the gap
  appear to radiate from the central axis of the
  cylinder and are always to the plane of the
  coil as it rotates.

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• Pointer-type galvanometer
• In a pointer-type galvanometer, the coil is
  pivoted on jewelled bearings. The rotation of
  the coil is resisted by above and below it.
  The springs also lead in and out of the coil.
  
• Light-beam galvanometer
• In a light-beam galvanometer, the coil is
  suspended and controlled by two gold alloy
  held taut by above and below it. The ribbons
  conduct to and from the coil. A small fixed
  to the coil throws the of an illuminated hair
  line onto the scale. The angular deflection of
  the coil is magnified by such optical
  system.

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• Theory of moving-coil galvanometer
• The field produces forces on the
  vertical sides which are always to
  the plane of the coil so that the couple has a
  value at all positions.
• If the air gap is of constant width, the flux
  density B of the field at the coil is also
  . The deflecting couple is given by ? .
• The coil rotates until the resisting torque ?
  due to the suspension is and to ?. If
  the deflection is then ? and k is the torque
  exerted by the hair spring per unit angular
  deflection, in , we have at
  equilibrium Eq.(3)

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• Sensitivity of moving-coil galvanometer
• The use of a field and a gap thus
  results in the equilibrium deflection ? of coil
  being to the current I.
• The current sensitivity of a galvanometer,
  defined as the thus equals
• Maximum current sensitivity therefore requires
• 1. B to be in the air gap, i.e., the magnets
  should be and the air gap should be
• 2. A to be , but not at the expense
  to make the coil
• about the equilibrium deflection before a
  reading can be taken
• 3. N to be but not at the expense of having
  to use a air gap
• 4. k to be , i.e., the hair springs should be
  but not at the expense to make
  readings take .

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Examination questions

• 1991-IIA-37
• It is desired to re-design a moving coil
  galvanometer so as to make it four times as
  sensitive. Which of the following would alone
  achieve the desired result?
• (1) Increasing the magnetic flux density of the
  permanent magnet to twice its value and doubling
  the cross-sectional area of the coil.
• (2) Providing the coil with a shunt so that only
  a quarter of the input current flows through the
  coil itself.
• (3) Changing the suspension characteristics so
  that four times as great a couple is needed to
  cause one radian twist.
• A. (1), (2) and (3) B. (1) and (2) only C.
  (2) and (3) only
• D. (1) only E. (3) only
• 1992-IIA-33
• A moving coil galvanometer with coil of area A
  and N turns has a full-scale deflection for a
  current i. If the coil were of area 3 A and 2 N
  turns, the current which would give full scale
  deflection would be
• A. i/6 B. 2i/3 C. i D.
  3i/2 E. 6i

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• 1992-IIA-34
• A square coil of N turns and area A carries a
  current I. The coil is free to rotate about the
  axis XY which is normal to a uniform magnetic
  field B. When the field makes an angle ? with
  the plane of coil, what are the magnitude and
  direction of the torque acting on the coil as
  detected by an observer at X?
•   Magnitude Direction
• A. BANIcos? clockwise
• B. BANIsin? clockwise
• C. BANIcos? anticlockwise
• D. BANIsin? anticlockwise
• E. None of the above

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• 1998-IIA-30
• The above diagram shows a rectangular
  current-carrying coil ABCD in a uniform magnetic
  field between two pole pieces. The magnetic
  field is perpendicular to the plane of the coil.
  Which of the following statements is/are correct?
• (1) There is a magnetic force acting on the side
  BC of the coil.
• (2) The magnetic forces acting on the coil tend
  to reduce its area.
• (3) The coil will return quickly to its original
  vertical position when it is disturbed slightly.
•   A. (1) only B. (3) only C. (1) and (2)
  only
• D. (2) and (3) only E. (1), (2) and (3)

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• 2001-IIA-29
• Which of the following descriptions about a
  moving-coil meter is/are correct?
• (1) It has a massive soft iron core to provide
  damping.
• (2) It has curved magnetic poles to assist in
  producing linear scale.
• (3) It has weak hair springs to increase the
  sensitivity.
• A. (1) only B. (3) only C. (1) and (2)
  only
• D.  (2) and (3) only E. (1), (2) and (3)