Chapter 9 Magnetic Fields: Il 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 B 2Tp so over a circle ot radIus p centered on the wire(see g1) ∮B·dl 2丌 Figure 9.1 positive direction for the integration path around a current 1
This is a general result for any closed curve c lc B dI=uo ote the right-hang screw rule here for the directions of B and i This can be generalized to the ampere s law or a volume distribution of current 欠B.dl=0sJ·da, where J is the current density through 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 is IXN
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P2 Figure 9-3 Long cylindrical conductor carrying a current 1
Outside the conductor: B is azimuthal, according to the circuital law 2丌pB=p401.→B 2丌 Inside the conductor 2丌pB=10(mp),→B=0Jp_A0p 2 2丌B 20+×10 10 102030405060 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 10A 2xpB=140N.→B= 2 7p 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 by the path d, so there exist non-azimuthal component of B. It is very small though Remark Although outside the toroidal coil B=0, An hold ≠
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