A Ring of Current: In Griffiths’ example 5.6 (p. 227), he determines the magnetic field at a point directly above the center of a circular loop (? = ??̂) of current to be, ?(?) = ?0 ? 2 ?2 (?2 + ?2) 3/2 ̂? (1) where ? is the radius of the current loop, and ? is the distance above the center.
(a) Simplify this result for the following two cases. For both cases, your results must still have ? in the answer.
i. ? ≫ ?
ii. ? ≪ ? (Note: You should use a Taylor series expansion for this one assuming that ? ? ≈ 0.)
iii. What happens to the direction of ?(?) when ? goes from just above zero to just below zero? (That is, does the sign of ? matter?)
(b) Now set up a Biot-Savart-style integral to determine the magnetic field due to this current-carrying ring at a location in the plane of the ring: ?(?) at ? = ? ̂?. You will not be able to solve this integral analytically, but you can simplify the integrand for the following two cases and then integrate to get the magnetic field. For both cases, your results must still have ? in the answer.
i. ? ≫ ? (Consider ? ? ≈ 0). Compare this result to what you found in part 2(a)i.
ii. ? ≪ ? (Consider ? ? ≈ 0). Compare this result to what you found in part 2(a)ii.
iii. What happens to the direction of ?(?) when ? goes from just in front of zero to just behind zero? (That is, does the sign of ? matter?)
In both cases, the sign of 'z' or 'x' does not matter, as the expression does not directly proportional to the odd powers of 'z' or 'x'.
Problem 2(a) is not given, so the comparison of the 1(b) [i] &[ii] part is left.
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