IV. Electromagnetic Induction

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IV. Electromagnetic Induction

• Further relations between electric and magnetic
  fields

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IV1 Faradays Law
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Main Topics

• Introduction into Electro-magnetism.
• Faradays Experiment.
• Moving Conductive Rod.
• Faradays Law.
• Lenzs Law.
• Examples

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Introduction into Electro-magnetism

• Many scientists in history were interested in
  relation between electric and magnetic fields.
  When it was known that electric currents produce
  magnetic fields and interact with them a natural
  question has appeared do also magnetic fields
  produce electric fields?
• Simple experiments somehow didnt work.

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Faradays Experiment I

• Michael Faraday (1791-1867) used two coils on a
  single toroidal core. He used a power-source to
  produce a current through the first coil and he
  connected galvanometer to the other coil. He
  probably was not the first one to find out that
  there was no current through the galvanometer,
  regardless on how strong the current was.

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Faradays Experiment II

• But he was the first who noticed that the
  galvanometer deflected strongly when the power
  source was switched on and it also deflected in
  the opposite direction when he opened the switch
  and disconnected the power source.
• He concluded that the galvanometer reacts to the
  changes of the magnetic field.

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Simple Demonstration I

• We can show the effect of electromagnetic
  induction and all its qualitative properties even
  more simply. We need is a permanent magnet and
  few loops of wire, connected to a galvanometer.
• If we move the magnet into the coil the
  galvanometer moves in one direction if we move it
  out the deflection direction changes.

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Simple Demonstration II

• If we make the experiment more accurately, taking
  into account which pole of the magnet is the
  north we find out that the current has such a
  direction that the field it produces goes against
  the changes of the external field we do by moving
  the magnet.
• We can also notice that it is sufficient to tilt
  the magnet and keep in the same distance.

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Moving Conductive Rod I

• Before we state the general law describing the
  effect it is useful to study one special case of
  a conductive rod of a length L moving
  perpendicularly to the field lines of a uniform
  magnetic field with a speed v.
• Let us expect positive free charge carriers in
  the rod. Since we force them to move in magnetic
  field, they experience Lorenz force.

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Moving Conductive Rod II

• The charges are free in the rod so they will move
  and charge one end of the rod positively.
• The positive charge will be missing on the other
  end so an electric field appears in the rod. Its
  direction is opposite the the Lorenz force, so it
  can be expected the charging will continue only
  until some equilibrium is reached.

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Moving Conductive Rod III

• As the charges are positive the equilibrium will
  be reached when the electric and magnetic forces
  are equal so the net force on the charges is zero
  and the charging of the ends thereby stops
• qvB qE qV/L ? V BvL
• This is valid regardless on the polarity or the
  magnitude of the free charge carriers.

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The Magnetic Flux I

• We have seen that movement of a conductive rod in
  magnetic field leads to induction of EMF in it.
  This was a special case of change of a new
  quantity the flux of the magnetic induction or
  shortly the magnetic flux.

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The Magnetic Flux II

• The magnetic flux is defined as
• d?mB.dA
• It represents amount of magnetic induction B
  which flows perpendicularly through a small
  surface dA, characterized by its outer normal
  vector.
• Remember what exactly the scalar product means!

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The Gauss Law in Magnetism

• The total magnetic flux through a closed surface
  is always equal to zero!
• This is equivalent to the fact that magnetic
  monopoles dont exist so the magnetic field is
  the dipole field and its field lines are always
  closed.
• Any field line which crosses any closed surface
  must cross it also in again in opposite sense.

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The Faradays Law I

• The general version of Faradays law of induction
  states that the magnitude of the induced EMF in
  some circuit is equal to the rate of the change
  of the magnetic flux through this circuit
• ? - d?m/dt
• The minus sign describes the orientation of the
  EMF. A special law deals with that.

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The Faradays Law II

• The magnetic flux is a scalar product of two
  vectors, the magnetic induction B and A the
  normal describing the surface of the circuit. So
  in principle three quantities can change
  independently to change the magnetic flux
• B this happens in transformers
• A e.g. in our example with the rod
• relative direction of A and B generators

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The Lenzs Law

• The Lenzs law deals with the orientation of the
  induced EMF. It states
• An induced EMF gives rise to a current whose
  magnetic field opposes the original change in
  flux.
• If the circuit is not closed and no current flows
  we can imagine the directions if it was closed.

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Moving Conductive Rod IV

• Lets illustrate Lenzs law on our moving rod.
  Now we move it perpendicularly to two parallel
  rails.
• If we connect the rails on the left, the flux
  grows since the area of the circuit grows. The
  current must be counterclockwise so the field
  produced by it points into the plane and thereby
  opposes the grow in flux.

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Moving Conductive Rod V

• If we connect the rails on the right, the flux
  decreases since the area of the circuit
  decreases. The current must be clockwise so the
  field produced by it points out of the plane and
  thereby opposes the decrease in flux.
• Both cases correspond to the orientation of the
  EMF we have found previously.

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Simple Demonstration III

• If we return to the demonstration with a
  permanent magnet and a galvanometer.
• From its deflection we can see what is the
  direction of the the currents in the case we
  approach the wire loop and the case we leave it.
  From this we can find which pole of the magnet is
  the north and verify if in the magnetic field of
  the Earth.

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Rotating Conductive Rod I

• A conductive rod L long, is rotating with the
  angular speed ? perpendicularly to a uniform
  magnetic field B.What is the EMF?
• The rod is mowing the field lines so there is
  EMF. But each little piece of the rod moves with
  different velocity. We can imagine the rod like
  many little batteries in series. So we just
  integrate their voltages.

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Moving Conductive Rod VI

• A QUIZ
• Do we have to do work on the conductive rod to
  move it in magnetic field?

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Moving Conductive Rod VII

• The answer is
• NO after the equilibrium is reached between
  electric and magnetic forces and net current
  doesnt flow!
• The situation will change when we bridge the
  rails by a resistor. WHY ?

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Homework

• Chapter 29 1, 3, 4, 5, 23, 24, 25

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Things to read and learn

• Chapter 29 1, 2, 3, 5
• Try to understand all the details of the scalar
  and vector product of two vectors!
• Try to understand the physical background and
  ideas. Physics is not just inserting numbers into
  formulas!

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The vector or cross product I

• Let ca.b
• Definition (components)

The magnitude c
Is the surface of a parallelepiped made by a,b.
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The vector or cross product II
The vector c is perpendicular to the plane made
by the vectors a and b and they have to form a
right-turning system.
?ijk 1 (even permutation), -1 (odd), 0 (eq.)

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The scalar or dot product

• Let ca.b
• Definition I. (components)

Definition II (projection of one vector
into the direction of the other one)

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Gauss Law in Magnetism

• The exact definition

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Rotating Conductive Rod

• At first we have to deal with the directions. If
  the field lines come out of the plane and the rod
  rotates in positive direction the center of
  rotation will be negative. dV in dr

• And total EMF