W9_parallel_resonance.eps

Week 8: Faraday’s Law and Induction

(Est 2/25-3/4)

• Suppose a conducting bar moves through a field at right angles to the field lines and the
  alignment of the bar. Magnetic forces quickly push charges to the two ends until an electric
  field is created thatbalancesthe electric force. The integral of this field is called amotional
  potential difference.


• Suppose now that a rectangular wire loop is pushedinto(or pulled out of) a uniform field that
  terminates at an edge (perhaps generated by a solenoid with a slot init). We note that the
  field now pushes charges around the loop in agreement with the motional potential difference
  and that the net magnetic force on the current carrying wireresiststhe push into (or pull out
  of) the field.


• We consider a conducting rod on rails as it slides through such a field. We can see that the
  induced/motional potential difference is equal to the time rate of change of the field times the
  area the field occupies within the rectangle.


• Time for our final Maxwell equation. If the magnetic field flux through an open surfaceS
  bounded by a closed curveCvaries in timeitinducesanelectric fielddynamically around the
  closed curve according toFaraday’s Law:
  ∮

C

E~·d~ℓ=−d

dt

∫

S/C

B~·ˆndA (547)

The integral on the left is theinduced voltagearound the curveC.

• In this equation the minus sign is calledLenz’s Lawand tells us that the induced voltage
  decreases around the loop in the direction such that a flow of positive charge in that direction
  (theinduced currentif the loop is a conducting pathway) willoppose the changein the varying
  flux. If the flux is decreasing it will generate a magnetic moment thatpoints in the direction
  that will increase it. If it is increasing it will generate a magnetic moment that points in the
  direction that will decrease it. This causes theoppositionto motion noted in the motional
  voltage problems above.


• The flux through a conducting loop is directly proportional to the current through the loop
  itself or to the current through nearby sources of magnetic field that produce the flux. The
  constant of proportionality in either case depends solely on thegeometryof the loop and
  source(s). That is, given a bunch of loops:
  φi=

∑

j 6 =i

MijIj+LiIi (548)

where theMijare called themutual inductancesbetween theith andjth loops andLiis the

self inductanceof theith loop.

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