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29.3 Lenz's Law

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					                    29.3 Lenz's Law
• Is a convenient alternative method for determining the
  direction of an induced emf or current
• is not independent – can be derived
  from Faraday's law
• due to the Russian scientist
  H.F.E Lenz (1804-1865)
Lenz's law states:
The direction of any magnetic induction effect is
such as to oppose the cause of the effect.
The “cause” may be changing flux
• through a stationary circuit due to a changing B-field
• due to motion of the conductors that make up the circuit
                         M. Moodley, 2009                  1
If the flux in a stationary circuit changes
• the induced current sets up a B-field of its own
• within the area bounded by the circuit, this field is
  opposite to the original field if the original field is
  increasing
• but is in the same direction as the original field if the
  latter is decreasing
            the induced current opposes the change in
            flux through the circuit (not the flux itself)




                           M. Moodley, 2009                   2
If the flux change is due to motion of conductors
• the direction of the induced current in the moving
  conductor is such that the direction of the B-field force
  on the conductor is opposite in direction to its motion
          the motion of the conductor, which caused
          the induced current, is opposed

In all cases:
the induced current tries to preserve the status quo
by opposing motion or a change of flux




                          M. Moodley, 2009                    3
In the slidewire generator, the
induced current in the loop
causes an additional B-field in
the area bounded by the loop.
The direction of the induced
current is counter-clockwise.




                                             The magnitude of the
                                             uniform B-field is
                                             increasing, and the
                                             resulting induced
                                             current is in the
                                             clockwise direction
                          M. Moodley, 2009                          4
M. Moodley, 2009   5
In which direction is the
current induced in the
loop for each situation?




                            M. Moodley, 2009   6
        29.4 Motional Electromotive Force
To understand the origin of the induced emf we consider
the magnetic forces on mobile charges in the conductor:
(lets look at a rod moving in a uniform B-field)
                      • a charge +q in the rod
                        experiences a magnetic force
                      • the magnetic force causes free
                        charges in the rod to move and
                        excess positive charges collect at
                        the upper end a and negative
                        charges at the lower end b
                                       this creates an electric
                                       field E within the rod,
                                       directed from a toward b 
                                       and an electric force in
                                       the same direction
                        M. Moodley, 2009                      7
• charge continues to accumulate at the ends until E
  becomes large enough for the downward E-force to
  cancel the upward B-force
                             (charges are in equilibrium)
• the magnitude of the potential difference
  is then given by (using         )


                          • with point a at a higher
                            potential than point b




                         M. Moodley, 2009               8
Now suppose the moving rod slides along a stationary U-
shaped conductor, forming a complete circuit
• no magnetic force acts on the charges in the stationary
  U-shaped conductor
• but the charge that was near points a and b redistributes
  itself along the stationary conductors, creating an E-field
  within them
• this E-field creates a current in the counter-clockwise
  direction




                         M. Moodley, 2009                  9
• the moving rod has become a source of electromotive
  force
• within it, charge moves from lower to higher potential
• and in the remainder of the circuit, charge moves from
  higher to lower potential
• this emf is called a motional electromotive force :

• corresponds to a force per unit charge of magnitude vB
  acting for a distance L along the moving rod

Can generalise this for a conductor of any shape moving
in any B-field: for an element dl of conductor we have

For any closed conducting loop, the total motional emf is
                                        ( an alternative formulation
                                          of Faraday's law )
                         M. Moodley, 2009                         10
Example: Calculating motional emf
Suppose the length L in the figure is 0.10 m, the velocity v is 2.5
m/s, the total resistance of the loop is 0.030 Ω , and B is 0.60 T.
Find E (the induced emf), the induced current, and the force
acting on the rod.




                            M. Moodley, 2009                    11
Example: The Faraday disk dynamo
A conducting disk with radius R, lies in the xy-plane and rotates
with constant angular velocity ! about the z-axis. The disk is in
a uniform, constant B-field parallel to the z-axis. Find the
induced emf between the centre and the rim of the disk.




                           M. Moodley, 2009                    12
            29.5 Induced Electric Fields
An induced emf also occurs when there is a changing flux
through a stationary conductor.
Consider the following situation of a long, thin solenoid
with cross-sectional area A and n turns per unit length:
                                           G measures current
                                           in the loop




                        M. Moodley, 2009                  13
• the current I in the windings set up a B-field along the
   solenoid axis with magnitude

• taking the area vector A to point in the same direction as
  B, the magnetic flux through the loop is

• if the solenoid current I changes with time, the flux
  changes and according to Faraday's law the induced emf
  in the loop is


• if the total resistance in the loop is R, the induced current
  in the loop I' is




                          M. Moodley, 2009                   14
But what force makes the charges move around the loop?
• it cannot be due to the B-field
• it is due to an induced E-field in the conductor caused
  by the changing magnetic flux
• this E-field is not conservative
• the line integral, representing the work done by the
  induced E-field per unit charge is equal to the induced
  emf:

• Faraday's law can therefore be restated as



• is only valid if the path around which we integrate is
  stationary                M. Moodley, 2009                15
Example: Induced electric fields
Suppose the long solenoid is wound with 500 turns per metre
and the current in the windings is increasing at the rate of 2
100 A/s. The cross-sectional area of the solenoid is 4.0 cm .

a) Find the magnitude of the induced emf in the wire loop
   outside the solenoid.
                                              b) Find the magnitude
                                                 of the induced
                                                 electric field within
                                                 the loop if its radius
                                                 is 2.0 cm.




                           M. Moodley, 2009                          16

				
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