Chapter 9 Magnetic Fields: II Ampere' s circuital law Example: Long Cylindrical Conductor Example: Toroidal Coil Example: Refraction of Lines of b at a Current sheet Short solenoid The curl of b
Chapter 9 Magnetic Fields:II ◼ Ampere’s Circuital Law ◼ Example:Long Cylindrical Conductor ◼ Example:Toroidal Coil ◼ Example:Refraction of Lines of B at a Current Sheet ◼ Short Solenoid ◼ The Curl of B
9. 1 Ampere's circuit Law It is known that near a long straight wire the magnetic induction is 1 B 2丌 P1 so over a circle of radius p centered on the wire(see g9-1) ∮B·dl 丌p=10 Figure 9-1 Positive direction for the ntegration path around a current L
T his is a general result for any closed curve C B Note the right-hang screw rule here for the directions of B and i This can be generalized to the ampere's law For a volume distribution of current B·dl=p0sJda, where j is the current densi ity thro h ugh any surface s bounded by the curve C See Fig 9-2 a
C B
In many cases the same current crosses the surface bounded by the curve C several times See Fig. 9-2(b)for a solenoid Then the total current isⅠ×N, C
In many cases the same current crosses the surface bounded by the curve C several times. See Fig. 9-2(b) for a solenoid. Then the total current is I×N
Example: Long Cylindrical Conductor a current I is uniformly distributed over its cross section with a density 丌R ure 9-3 Long cylindrical conductor carrying a current
Outside the conductor: B is azimuthal, according to the circuital law 27B=10l.→B=A0 丌p Inside the conductor EpB=40J(p2),>B-koJpuolp 2丌7 20r×10 0 02030405060 p(millimeters) Figure 9-4 The magnetic induction B as a function of radius for a wire of I millimeter radius carrying a current of 1 ampere
Example: Toroidal Coil(see Fig 9-5) Inside: along the path b, one has the azimuthal com ponent 2丌pB =10NⅠ.→B=10r 27p Figure 9-5 Toroidal coil. The broken lines show paths of integration
Outside The azimuthal component of b is zero, as in the paths a and c But the current crosses once the area bounded the path d, so there exist non - azimuthal component of B. It is very small though Remark: Although outside the toroidal coil B=0 A uoIdI ≠0
Example: Long solonoid (see Figs. 9-6 and 9-7) It is very long so that the end effects will negligible Choose a cylindrical coordinates Figure 9-7 Components of B at P, in cylindrical coordinates
Example: Long Solonoid (see Figs. 9-6 and 9-7） It is very long so that the end effects will negligible. Choose a cylindrical coordinates