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 r dI x r = + µ 0 2 2 2 3 2 2 e j where dI dQ t dQ = = ω π 2 dQ dV rdr dx = = ρ ρ π 2 b ga f dB r drdx x r = + µ ρω 0 3 2 2 3 2 2 e j where ρ π = Q R 4 3 3 b g B r drdx x r r R x x R R = + = =− + z z µ ρω 0 3 2 2 3 2 0 2 2 2 e j r x dr dx ω R FIG. P30.71 Let v r x = + 2 2 , dv rdr = 2 , and r v x 2 2 = B v x dv v dx v dv x v dv dx B v x v dx R x x R x dx B x R x R v x R x R R v x R v x R x R R x R x R x R R x 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 z z z z z z µ ρω µ ρω µ ρω µ ρω µ ρω 0 2 3 2 0 1 2 2 3 2 0 1 2 2 1 2 0 2 0 2 2 2 4 4 2 2 4 2 2 1 1 4 2 4 2 2 2 2 2 2 2 2 2 2 2 e j e j c h O Q P = + L N M O Q P = + F H G I K J = z z dx x R x R dx B R R R R R R R R 2 4 2 4 2 2 4 2 3 4 2 2 3 0 2 0 0 3 2 2 0 2 µ ρω µ ρω µ ρω 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 = = ω π ρ π 2 2 b ga f . The magnetic moment contributed by this ring is
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