Consider a circular loop of radius R on the xy-plane centered at the origin, and the...

 

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Consider a circular loop of radius R on the xy-plane centered at the origin, and the current on the loop is I (constant). Because of the cylindrical symmetry of the current loop, the vector potential can be written as A(r) = A(s, z) $. a) Calculate A(s, z). You will get a very complicated integral. Estimate this integral for the limits s < R and s> R using suitable Taylor expansions. Hint: Notice that A(s, z) is independent of p. To find A(s, z), you can simply consider the y-component of A(r) at = 0. b) For s << R limit, calculate the magnetic field by B = VX A. Show that for s = 0 (i.e. on the z axis), we get the same result as in Griffiths' Example 5.6. c) Show that for s>> R limit, we get the dipole potential. Find the corresponding magnetic dipole moment. Consider a circular loop of radius R on the xy-plane centered at the origin, and the current on the loop is I (constant). Because of the cylindrical symmetry of the current loop, the vector potential can be written as A(r) = A(s, z) $. a) Calculate A(s, z). You will get a very complicated integral. Estimate this integral for the limits s < R and s> R using suitable Taylor expansions. Hint: Notice that A(s, z) is independent of p. To find A(s, z), you can simply consider the y-component of A(r) at = 0. b) For s << R limit, calculate the magnetic field by B = VX A. Show that for s = 0 (i.e. on the z axis), we get the same result as in Griffiths' Example 5.6. c) Show that for s>> R limit, we get the dipole potential. Find the corresponding magnetic dipole moment.