870_Physics ProblemsTechnical Physics

870_Physics ProblemsTechnical Physics - 210 P30.71 Sources...

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210 Sources of the Magnetic Field P30.71 Consider the sphere as being built up of little rings of radius r , centered on the rotation axis. The contribution to the field from each ring is dB rd I xr = + µ 0 2 22 32 2 ej where dI dQ t dQ == ω π 2 dQ dV rdr dx ρρ 2 bg af dB r d x = + µρω 0 3 2 where ρ = Q R 43 3 B r d x r Rx xR R = + = =− + z z 0 3 0 2 r x dr dx ω R FIG. P30.71 Let vr x =+ , dv rdr = 2 , and rv x B vxd v v dx v dv x v dv dx Bv x v d x R x x dx B x R vx R R R R R x R x R R R = L N M M O Q P P L N M O Q P + F H G I K J L N M M O Q P P + L N M = + −− = = z zz z z 0 2 0 12 2 0 2 0 2 0 2 2 2 4 4 4 11 4 24 2 2 2 2 2 2 2 2 2 2 2 ch O Q P + L N M O Q P + F H G I K J = dx x R d x B R R R R R R RR 2 4 2 2 4 2 3 4 2 2 3 0 2 0 0 2 0 2 P30.72 Consider the sphere as being built up of little rings of radius r , centered on the rotation axis. The current associated with each rotating ring of charge is dI dQ t rdr dx ρπ 2 2 a f . The magnetic moment contributed by this ring is d A dI r rdr dx r drdx µπ πωρ = 23 2 2 r x dr dx ω R FIG. P30.72 = L N M M O Q P P = F H I K = + = F H G I K J + L N M M O Q P P + F H G I K J = F H G I K J = = + + + + z z z x dx
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