F10HW10 - 1 Problem 5.21 1.1 ~~ B ¡ Hd3 x = = =  Z Z  ~~...

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Unformatted text preview: 1 Problem 5.21 1.1 ~~ B ¡ Hd3 x = = =  Z Z  ~~ r ¢ A ¡ Hd3x ~ A¡  ~ r¢H  d3 x ~J ~ A ¡ |{z} d3 x = 0 0 1.2 Starting from equation 5.72 in Jackson: W = U = m ¡ B ~~ For a discrete number of point dipoles, we can write: X 1X m ¡B ~~ W = mi ¡ Bj = ~~ 2 i6=j i j i<j Hence, for the continuous case, we can write: Z 1 ~~ M ¡ Bd3 x W = 2Z h i 1 ~ ~~ = M ¡ 0 H + M d3 x 2Z Z = 0  ~~ M ¡ Hd3 x 0 ~~ M ¡ Md3 x 2 2 Note that the second integral is a constant which is independent of the position or orientation of the magnetized bodies (since we're integrating over all space). ~ Plugging in H = ~ 0 B 1 ~ M into the above equation yields: Z   Z  1~ ~ ~ ~~ W = M¡ B M d3 x 0 M ¡ Md3 x 2 0 2 Z ¨0 B 1 ~ ¨~ ¨ 0 ¨ 3x + Z M ¡ Md3x 0 Z M ¡ Md3x ~~ ~~ = M ¡ Bd 2Z ¨¨ 2Z 2 ¨ 0 = 0  ~~ M ¡ Md3 x 0 ~~ M ¡ Md3 x 2 2 where we've used the result from the previous part to show that the integral vanishes in the second step. 1 2 Problem 5.27 Let's pick our orientation such that the current through the inner wire is in the z direction: ^ ~^ J =z ( I b2 r<b r>b 0 The current enclosed a loop of radius r < b is: I = 2 Z r 0 Ir2 I rdr = 2 b2 b ~ Using Ampre's Law, we can nd the B eld at all points in space: e 8 Ir > 2b2 r < b < ~ = ' 0 I b < r < a B ^ 2r > :0 r>a Finally, we use equation 5.157 in Jackson to nd the self inductance: Z~~ B¡B 3 L= 2 dx I  # "Z   Z 1b Ir 2 1 a 0 I 2 L1 rdr = 2 rdr + ` I2  r=0 2b2 0 r=b 2r 2  L1 I 1 4 0 I 2  a  = 2 b + 2 ln ` I2 4 2 b4 4 4 b L  0  a  1 ` = + 8 2 ln b If the inner conductor is a thin hollow tube, all the current will ow on the outside of the ~ ~ tube. Hence, there is no current (and hence no B eld) for r < b. That is, B simpli es to: ~^ B=' ( I 0 2r 0 b<r<a otherwise And Equation 5.157 in Jackson becomes: Z~~ B¡B 3 L= 2 dx I "  Z  2 #  2   L1 1 a 0 I 1 I a = 2 2 rdr = 2 2 0 2 ln `I 0 r=b 2r I 4 b L 0  a  1 ` = 2 2 ln b ...
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This note was uploaded on 12/08/2011 for the course PHYSICS 505 taught by Professor Liu during the Winter '08 term at University of Michigan.

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F10HW10 - 1 Problem 5.21 1.1 ~~ B ¡ Hd3 x = = =  Z Z  ~~...

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