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Unformatted text preview: Two Ways of Getting B-Field of Magnetic Dipole
As I implied in class, there are two di erent ways of getting the magnetic induction B for a magnetic dipole m. The rst is that speci cally discussed in Jackson. It involves writing B=r A
and using the result (for a magnetic dipole moment at the origin) where I have written (1) (2) (3) A(x) = 4 o mr3 x ; If one carries out the curl operation, one nds that B(x) takes the form rr ^ B(x) = 4 o 3^(^ m) ? m : (4) r3 This form for the magnetic induction is accurate so long as r a, where a is a typical linear dimension of the magnetic dipole. The magnetic induction B of a magnetic dipole can also be obtained from the magnetic scalar potential. Outside the dipole, there is no free current density, so r H = 0. Also, if this is a region of non-magnetic material, ? B = 0H, and therefore r H = 0 1r B = 0. Therefore, H can be written as the gradient of a scalar potential, x = r^: r H = ?r
where (5) (6) The potential should be azimuthally symmetric with respect to m. Therefore, the leading term at large distances must have the form = C mr3 x ; 1 (7) r2 = 0: where C is a constant to be determined. Therefore, B has the form rr B = ? r = C 3^(^ m) ? m :
0 0 Comparing eqs. (8) and (4), we see that they are consistent only if C = 41 (9) and therefore (10) = 41 m 3 x : r In summary, one can obtain the magnetic induction B of a magnetic dipole m either as B = r A, with A given by eq. (2), or as B = ? 0r , with given by eq. (11). Note that when one uses the scalar potential method, the scalar potential for H has the same form as that for the electric eld E of an electric dipole, provided that one makes the replacements p ! m and 1=(4 0) ! 1=(4 ). r3 (8) 2 ...
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This note was uploaded on 07/17/2008 for the course PHYS 834 taught by Professor Stroud during the Fall '04 term at Ohio State.
- Fall '04