1. Phy 213: General Physics III Chapter 30: Induction & Inductance Lecture Notes

2. Electromagnetic Induction We have observed that force is exerted on a charge by either and E field or a B field (when charge is moving): Consequences of the Lorentz Force: –A B field can exert a force on an electric current (moving charge) –A changing B-field (such as a moving magnet) will exert a magnetic force on a static charge, producing an electric current → this is called electromagnetic induction Faraday’s contribution to this observation: –For a closed loop, a current is induced when: 1.The B-field through the loop changes 2.The area (A) of the loop changes 3.The orientation of B and A changes q N S q N S A current is induced ONLY when any or all of the above are changing The magnitude of the induced current depends on the rate of change of 1-3 Moving charge Moving magnet

3. Magnetic Flux Faraday referred to changes in B field, area and orientation as changes in magnetic flux inside the closed loop The formal definition of magnetic flux (  B  (analogous to electric flux)  When B is uniform over A, this becomes: Magnetic flux is a measure of the # of B field lines within a closed area (or in this case a loop or coil of wire) Changes in B, A and/or  change the magnetic flux Faraday’s Law: changing magnetic flux induces electromotive force (& thus current) in a closed wire loop 

4. Faraday’s Law When no voltage source is present, current will flow around a closed loop or coil when an electric field is present parallel to the current flow. Charge flows due to the presence of electromotive force, or emf (  ) on charge carriers in the coil. The emf is given by: An E-field is induced along a coil when the magnetic flux changes, producing an emf (  ). The induced emf is related to: –The number of loops (N) in the coil –The rate at which the magnetic flux is changing inside the loop(s), or Note: magnetic flux changes when either the magnetic field (B), the area (A) or the orientation (cos  ) of the loop changes: i

5. Changing Magnetic Field A magnet moves toward a loop of wire (N=10 & A is 0.02 m 2 ). During the movement, B changes from is 0.0 T to 1.5 T in 3 s (R loop is 2  ). 1)What is the induced  in the loop? 2)What is the induced current in the loop?

6. Changing Area A loop of wire (N=10) contracts from 0.03 m 2 to 0.01 m 2 in 0.5 s, where B is 0.5 T and  is 0 o (R loop is 1  ). 1)What is the induced  in the loop? 2)What is the induced current in the loop?

7. Changing Orientation A loop of wire (N=10) rotates from 0 o to 90 o in 1.5 s, B is 0.5 T and A is 0.02 m 2 (R loop is 2  ). 1)What is the average angular frequency,  ? 2)What is the induced  in the loop? 3)What is the induced current in the loop?

8. Lenz’s Law When the magnetic flux changes within a loop of wire, the induced current resists the changing flux The direction of the induced current always produces a magnetic field that resists the change in magnetic flux (blue arrows) Review the previous examples and determine the direction of the current Magnetic flux,  B Increasing  B i i

9. Operating a light bulb with motional EMF Consider a rectangular loop placed within a magnetic field, with a moveable rail (R loop = 2  ). B = 0.5 T v = 10 m/s L = 1.0 m Questions: 1) What is the area of the loop? 2) How does the area vary with v? 3) What is the induced  in the loop? 4) What is the induced current in the loop? 5) What is the direction of the current?

10. Force & Magnetic Induction What about the force applied by the hand to keep the rail moving? The moving rail induces an electric current and also produces power to drive the current: P = . i = (5 V)(2.5 A) = 12.5 W The power (rate of work performed) comes from the effort of the hand to push the rail –Since v is constant, the magnetic field exerts a resistive force on the rail: The force of the hand can be determined from the power:

11. Generators & Alternating Current Generators are devices that utilize electromagnetic induction to produce electricity Generators convert mechanical energy into electrical energy –Mechanical energy is utilized to either: Rotate a magnet inside a wire coil Rotate a wire coil inside a magnetic field –In both cases, the magnetic flux inside the coil changes producing an induced voltage –As the magnet or coil rotates, it produces an alternating current (AC) {due to the changing orientation of the coil and the magnetic field} Motors and Generators are equivalent devices –A generator is a motor running in reverse:

12. Maxwell’s Equations Taken in combination, the electromagnetic equations are referred to as Maxwell’s Equations: 1.Gauss’ Law (E) 2.Gauss’ Law (B) 3.Ampere’s Law 4.Faraday’s Law

13. Significance of Maxwell’s Equations 1.A time changing E field induces a B field. 2.A time changing B field induces an E field. 3.Together, 1 & 2 explain all electromagnetic behavior (in a classical sense) AND suggest that both E & B propagate as traveling waves, directed perpendicular to each other AND the propagation of the waves, where: and The product,  o  o, has special significance: or