ch32-p058 - 58. (a) At a distance r from the center of the...

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where µ is the Earth’s dipole moment and λ m is the magnetic latitude. The ratio of the field magnitudes for two different distances at the same latitude is B B r r 2 1 1 3 2 3 = . With B 1 being the value at the surface and B 2 being half of B 1 , we set r 1 equal to the radius R e of the Earth and r 2 equal to R e + h , where h is altitude at which B is half its value at the surface. Thus, 1 2 3 3 = + R Rh e e b g . Taking the cube root of both sides and solving for h , we get () ( )( ) 13 3 2 1 2 1 6370km 1.66 10 km. e hR =−=− (b) We use the expression for B obtained in Problem 32-55, part (a). For maximum B , we set sin λ m = 1.00. Also, r = 6370 km – 2900 km = 3470 km. Thus, ( ) ( ) 72 2 2 2 2 0 max 3 3 6 4 4 10 T m A 8.00 10 A m s i n 131 . 0 0 4 4m 3.83 10 T. m B r µµ π× × =+ λ = + π π3 .47×10 (c) The angle between the magnetic axis and the rotational axis of the Earth is 11.5°, so λ m = 90.0° – 11.5° = 78.5° at Earth’s geographic north pole. Also r = R e = 6370 km. Thus,
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