Chapter 29

Electromagnetic Induction We know Current loop + magnetic field  torque But does a Torque + magnetic field  current?

Faraday’s Law The circle represents the current in the loop. How to get more current Use two magnets Make loop larger Make magnet faster So the more field lines which move through the coil are what causes the current Recall

Faraday’s Law Cont’d The induced EMF in a closed loop equals the negative time-rate of change of the magnetic field. But what is EMF?

EMF—Electromotive Force Any chemical, solar, mechanical, heat method of creating a potential difference Batteries Alternator, dynamo Solarcell, photovoltaic Thermocouple Symbol E Units: volts

But what is the direction? Binduced Binduced The circle represents the current in the loop. Lenz’s Law: The direction of any magnetic induction effect is such as to oppose the cause of the effect.

Mathematically, The magnetic flux is the sum of the magnetic field lines going through an area. It changes with time if: The magnetic field varies with time The area of the loop varies with time The angle that B makes with the normal of the surface of dA varies with time The “minus” sign is to show that the EMF acts to oppose the magnetic field generating it. SI unit for magnetic flux is weber = 1 T*m2

3 Ways to Change the Flux Move the Magnet Grow/Shrink the Ring Spin the Magnet or Spin the coil

Faraday’s Law Direction Rules Define the normal to the area under consideration as positive (A is positive) From Faraday’s Law, find the sign of the flux, F, and the sign of its rate of change, d F /dt If flux increasing, d F /dt>0 If flux decreasing, d F /dt<0 Since EMF=-(d F /dt), then d F /dt>0 implies EMF<0 d F /dt>0 implies EMF>0 Point thumb of right hand In negative direction if EMF<0 In positive direction if EMF>0 Your fingers of your RH are now pointing in the direction of current.

Electric Guitar Induced Magnetic Field on String String S N S V N Permanent Magnet

Slide Wire Generator v Area=x*L But X X X X X X X X X X X X X X v=dx/dt d(area)/dt=vL X X X X X X X X X X X X X X L R EMF=dF/dt=B*d(area)/dt EMF=BvL B into page EMF=iR i=EMF/R = BvL/R P=(EMF)*I P=B2v2L2/R

Hey, wait a minute, we created a current, so that must mean there is now an electric field! But the induced current is moving in a circle so So a time varying magnetic creates an electric field with no divergence

But earlier, you said Yes, but that was for ELECTROSTATIC situations Now the charges are moving! Electric potential has meaning only for electric fields produced by static charges. It has no meaning for electric fields produced by induction

Recall So it should surprise you that the closed path integral of the magnetic field is Note the difference in signs so for similar situations, the curl of the B field will be opposite the curl of the E field In order to keep things straight in his own mind, Maxwell called this a “displacement current”. He could then use Ampere’s law

Displacement Current Maxwell’s fictional current -q +q i i id This plate induces a negative charge here Which means the positive charge carriers are moving here and thus a positive current moving to the right

Maxwell’s Equations The electric field spreads into space proportional to the amount of static charge and how closely you space the static charges Magnetic field lines are closed loops and always return to the source creating them An electric field, resembling a magnetic field in shape, can be created by a time-varying magnetic field. There are two ways to produce a magnetic field: 1) by a current and 2) by a time-varying electric field.