Chapter 1l Magnetic Fields: IV Motional electromotance Faraday induction Law forv x B Fields Lenz' law Faraday Induction Law for Time Dependent B Flux Linkage E in terms of∨andA
Chapter 11 Magnetic Fields:IV ◼ Motional Electromtance ◼ Faraday Induction Law for v x B Fields ◼ Lenz’ Law ◼ Faraday Induction Law for TimeDependenct B ◼ Flux Linkage ◼ E in Terms of V and A
In this chapter we are with two phenomena 1)The Lorentz force Qv b on the charge carriers inside a moving conductor If a magnetic field is time-dependent then there ppears an electric field oA(t/at
11.1 Motional electromotance Consider a conductor moving at a velocity v in a magnetic field The conduction electrons inside the conductor also move with v Then we know that the conduction electrons drifts driven by the lorentz force - ev X B If the conductor forms a closed circuit c. then the electrons move forming a current in the circuit as if there were a battery supplying a voltage =(v×B)
Remarks 1)v is called the induced electromotance or the motional electromotance. lts unit is volt 2)v adds algebraically to the voltages of other sources hat may be present in the circuit
11.2 The Faraday Induction Law for vX B Fields The induced electromotance can be written as =(v×B)·dl=-kB·(v×dl, Where we have used the formula (A×B)C=-B·(A×C), which is true for any vectors a, b and c
Consider Fig 11-1 The element dl moves at the velocity v The product dr da (v×dl)= dt dl=(dr×d)lt dt where da- drx di is the area swept by the element di over a small displacement dr
Thus V=frda d更 dt dt where gp is the magnetic flux passing through the closed path C, and d is its variance caused by the displacement dr The direction of y is determined as follows drives the current that generates a magnetic field in the opposite direction to the increase of the orig nal B The Faraday Law for vX B field
Example: An Expanding Loop(Fig 11-2 B points into the page, v points to the right hand side, and the vector vxB points upward in the same direction as dl. so the induced electromotance is =f(v×B).dl=vB
Example: Loop Rotating in a Magnetic Field When the normal of the loop plane is has an angle 8 with B, the flux through the loop is =/B. da=/ Bcos e da= B cos 0( d abBa sin(w dt
113Lenz’LaW As we have already pointed out that from d dt one determines the direction of y If the magnetic flux linkage increases, do/dt is pos itive,and v is in the opposite direction Thus the induced current is such that its magnetic fields opposes the change in flux -The lenz law