Physics 121: Electricity

1
Physics 121 Electricity Magnetism Lecture
11Induction I

• Dale E. Gary
• Wenda Cao
• NJIT Physics Department

2
Currents Create Magnetic Fields

• B due to long straight wire carrying a current i
  
• B due to complete loop carrying a current i
• B inside a solenoid a torus
  carrying a current i

3
Induced Emf and Current

• A wire of length l is moving through a uniform
  magnetic field directed into the board.
• Moving in a direction perpendicular to the field
  with constant velocity v.
• Electrons feel a magnetic force and migrate,
  producing an induced electric field E.
• Charges come to equilibrium when the forces on
  charges balance
• Electric field is related to potential difference
  across the ends of wire
• A potential difference is maintained between the
  ends of the wire as long as the wire continues to
  move through the magnetic field.

• A current is set up even through no batteries are
  present in the circuit.
• Such a current is an induced current.
• It is produced by an induced emf.

4
Faradays Law Experiments

• A current appears only if there is relative
  motion between the loop and the magnet the
  current disappears when the relative motion
  between them ceases.
• Faster motion produces a greater current.
• If moving the magnets north pole toward the loop
  causes, say, clockwise current, then moving the
  north pole away causes counterclockwise current.
  Moving the south pole toward or away from the
  loop also causes currents, but in the reversed
  directions.

• An emf is induced in the loop when the number of
  magnetic field lines that pass through the loop
  is changing.

5
Flux of Magnetic Field

• We need a way to calculate the amount of magnetic
  field that passes through a loop.
• Similar to the definition of electric flux, we
  define a magnetic flux
• Magnetic flux is a scalar.
• In uniform magnetic field, the magnetic flux can
  be expressed as
• SI unit is the weber (Wb)
• 1 weber 1 Wb 1 T m2

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Faradays Law of Induction

• The magnitude of the emf induced in a conducting
  loop is equal to the rate at which the magnetic
  flux through that loop changes with time,
• If a coil consists of N loops with the same area,
• the total induced emf in the coil is given by
• In uniform magnetic field, the induced emf can be
  
• expressed as
• Emf can be induced in several ways,
• The magnitude of B can change with time.
• The area enclosed by the loop can change with
  time.
• The angle between B and the normal to the loop
  can change with time.
• Any combination of the above can occur.

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Induced Current and Emf

• A circular loop of wire is held in a uniform
  magnetic field, with the plane of the loop
  perpendicular to the field lines. Which of the
  following will not cause a current to be induced
  in the loop?
• Pushing the loop into the field.
• Rotating the loop about an axis perpendicular to
  the field lines.
• Keeping the orientation of the loop fixed and
  moving it along the field lines.
• Crushing the loop.
• Pulling the loop out of the field.

B
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Induced Current and Emf
2. The graph gives the magnitude B(t) of a
uniform magnetic field that exists throughout a
conducting loop, with the direction of the field
perpendicular to the plane of the loop. In which
region of the graph, the magnitude of the induced
emf is the greatest?
B(t)
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Induction and Energy Transfers

• A conducting bar of length l sliding along two
  fixed parallel conducting rails.
• Free charges feel a magnetic force along the
  length of the bar, producing an induced current
  I.
• Start with magnetic flux
• Follow Faradays law, we have
• Then
• Origin of the induced current and the energy
  dissipated by the resistor?

• The change in energy in the system must equal to
  the transfer of energy into the system by work.
• Moving with constant velocity,
• Power by the applied force is

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Lenzs Law

• Lenzs law for determining the direction of an
  induced current in a loop.
• The induced current in a loop is in the direction
  that creates a magnetic field that opposes the
  change in magnetic flux through the area enclosed
  by the loop.
• The direction of an induced emf is that of the
  induced current.
• The induced current tends to keep the original
  magnetic flux through the loop from changing.

• Work by external agent induces current.
• Induced Bi does not always opposes B.

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Direction of induced current

• Which figure is physically reasonable?

A
C
B
v
v
v0
i
i
i
D
E
v
v
i
i
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Direction of induced current

• 4 A circular loop of wire falling toward a
  wire carrying a current to the left. What is the
  direction of the induced current in the loop of
  wire?
• Clockwise
• Counterclockwise
• Zero
• Impossible to determine

v
I
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A Loop Moving Through a Magnetic Field

• A rectangular metallic loop of dimensions l and w
  and resistance R moves with constant speed v to
  the right. It passes through a uniform magnetic
  field B directed into the page and extending a
  distance 3w along the x axis. Define x as the
  position of the right side of the loop along the
  x axis.
• Plot as a function of x the magnetic flux, the
  induced emf, the external applied force necessary
  to keep v constant.
• Definitions
• Before entering field
• Entering field
• Entirely in field
• Leaving field
• After leaving field

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Induced Electric Fields

• A uniform field fills a cylindrical volume of
  radius R. Suppose that we increase the strength
  of this field at a steady rate by increasing.
• Copper ring A changing magnetic field produces
  an electric field.
• By Faradays law, an induced emf and current will
  appear in the ring
• From Lenzs law, the current flow
  counterclockwise
• An induced electric field must be present along
  the ring
• The existence of an electric field is independent
  of the presence of any test charges. Even in the
  absence of the copper ring, a changing magnetic
  field generates an electric field in empty space.
• Hypothetical circle path the electric field
  induced at various points around the circle path
  must be tangent to the circle.
• The electric field lines produced by the changing
  magnetic field must be a set of concentric
  circles.
• A changing magnetic field produces an electric
  field.

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A Reformulation of Faradays Law

• A charge q0 moving around the circular path.
• The work W done by the induced electric field,
• The work done in moving the test charge around
  the path,
• Two expressions for W equal to each other, we
  find,
• A more general expression for the work done on a
  charge q0 moving along any closed path,
• So,
• Combined with Faradays law,
• Electric potential has meaning only for electric
  fields produced by static charges it has no
  meaning for that by induction.

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Find Induced Electric Field

• In the right figure, dB/dt constant, find the
  expression for the magnitude E of the induced
  electric field at points within and outside the
  magnetic field.
• Due to symmetry,
• r lt R
• So,
• r gt R
• So,
• The magnitude of electric field induced inside
  the magnetic field increases linearly with r.

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Magnetic Field and Electric Field

• 5. The figure shows five lettered regions in
  which a uniform magnetic field extends either
  directly out of the page or into the page, with
  the direction indicated only for region a. The
  field is increasing in magnitude at the same
  steady rate in all five regions the regions are
  identical in area. Also shown as four numbered
  paths along which has the magnitudes
  given below. Determine the directions of magnetic
  field.
• b c d e
• b c d e
• b c d e
• b c d e
• b c d e

Path 1 2 3 4
mag 2(mag) 3(mag) 0
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Summary

• The magnetic flux ?B through an area A in a
  magnetic field B is defined as
• The SI unit of magnetic flux is the weber (Wb)
  1Wb 1Tm2.
• If the magnetic flux ?B through an area bounded
  by a closed conducting loop changes with time, a
  current and an emf are produced in the loop this
  process is called induction. The induced emf is
• If the loop is replaced by a closely packed coil
  of N turns, the induced emf is
• An induced current has a direction such that the
  magnetic field due to the current opposes the
  change in the magnetic flux that induces the
  current. The induce emf has the same direction as
  the induce current.
• An emf is induce by a changing magnetic flux even
  if the loop through which the flux is changing is
  not a physical conductor but an imaginary line.
  The changing magnetic field induces an electric
  field E at every point of such a loop the
  induced emf is related to E by
• where the integration is taken around the
  loop. We can write Faradays law in its most
  general form,
• The essence of this law is that a changing
  magnetic field induces an electric field E.