P32_008 - r = R e = 6370 km. Thus, B = 4 R 3 E q 1 + 3 sin...

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8. (a) At a distance r from the center of the Earth, the magnitude of the magnetic Feld is given by B = µ 0 µ 4 πr 3 q 1 + 3 sin 2 λ m , where µ is the Earth’s dipole moment and λ m is the magnetic latitude. The ratio of the Feld magnitudes for two di±erent distances at the same latitude is B 2 B 1 = r 3 1 r 3 2 . With B 1 being the value at the surface and B 2 being half of B 1 ,weset 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 = R 3 e ( R e + h ) 3 . Taking the cube root of both sides and solving for h ,weget h = ³ 2 1 / 3 1 ´ R e = ³ 2 1 / 3 1 ´ (6370 km) = 1660 km . (b) We use the expression for B obtained in problem 6, part (a). ²or maximum B , we set sin λ m =1 . Also, r = 6370 km 2900 km = 3470 km. Thus, B max = µ 0 µ 4 πr 3 q 1 + 3 sin 2 λ m = ( 4 π × 10 7 T · m / A ) (8 . 00 × 10 22 A · m 2 ) 4 π (3 . 47 × 10 6 m) 3 p 1+3(1) 2 =3 . 83 × 10 4 T . (c) The angle between the magnetic axis and the rotational axis of the Earth is 11 . 5 ,s o λ m = 90 . 0 11 . 5 =78 . 5
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Unformatted text preview: r = R e = 6370 km. Thus, B = 4 R 3 E q 1 + 3 sin 2 m = ( 4 10 7 T m / A ) (8 . 10 22 J / T) p 1 + 3 sin 2 78 . 5 4 (6 . 37 10 6 m) 3 = 6 . 11 10 5 T , and, using the result of part (b) of problem 6, i = tan 1 (2 tan 78 . 5 ) = 84 . 2 . A plausible explanation to the discrepancy between the calculated and measured values of the Earths magnetic Feld is that the formulas we obtained in problem 6 are based on dipole approx-imation, which does not accurately represent the Earths actual magnetic Feld distribution on or near its surface. (Incidentally, the dipole approximation becomes more reliable when we calculate the Earths magnetic Feld far from its center.)...
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