Chapter 11 Magnetic Fields: IV Motional electromotance a faraday induction Law forVx B Fields a 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 x b on the charge carriers inside a moving conductor 2)If a magnetic field is time-dependent, then there appears an electric field -aA(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 =f(v×B)
Remarks 1v is called the induced electromotance, or the motional electromotance. Its unit is volt 2 )V adds algebraically to the voltages of other sources that may be present in the circuit
11.2 The Faraday Induction Law for v B Fields The induced electromotance can be written as =(v×B)·dl=-B·(v×dl, where we have used the formul (A×B)C=-B·(AxC), which is true for any vectors a, b, and c
Consider fig 11-1 The element dl moves at the velocity v The product r da dt dl=(dr×dlt dt where da= dr x dl is the area swept by the element dl over a small displacement dr
Thus da d更 =-B dt dt where qp is the magnetic flux passing through the closed path C, and dq is its variance caused by the displacement dr The direction of y is determined as follows V drives the current that generates a magnetic field in the opposite direction to the increase of the orig- inal B 一 The Faraday Law for v× 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 =(v×B)·.dl=vBl. x B
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 e(ab) dΦ d abBa sinu
11.3 Lenz law As we have already pointed out that from d dt one determines the direction of y If the magnetic flux linkage increases, d/dt is pos- itive and v is in the opposite dir rection Thus the induced current is such that its magnetic welds opposes th e change in flux The lenz’LaW
