Recall from your course in physics that the motion of an electric charge around

Question

Recall from your course in physics that the motion of an electric charge around a closed loop produces a magnetic dipole, ?, whose direction is perpendicular to the loop and whose magnitude is given by ? = iA; where i is the current in amperes and A is the area of the loop (m^2). Show that i = qv/2?r for a circular loop, where v is the velocity of the charge

q and r is the radius of the loop. Show that ? = qrv/2 for a circular loop. If the loop is circular then we must use vector calculus and the magnetic dipole is given by ? = q(rv)/2. Show that this formula reduces to the preceding one for a circular loop. Last, using the relationship L = rp, show that ? = q/2m(L). Thus, the orbital motion of an electron in an atom imparts a magnetic moment to the atom. For an electron q = -|e| and so ? = (L)[|e|/2m(electron)].

Explanation / Answer

a) for a circular loop

the current is given by i = q/t

but the t here will be the time period of revolution

and hence t = 2*pi*r/v

so i =    (qv/2pi r)   

b) now the magnetic diipole is given by

M = i*A

=    (rac{qv}{2pi r}*pi r^2 = qvr/2)   

c) L = rp

or L = mvr

so vr = L/m

so magnetic dipole M = q/2 *rv = q/2 *(L/m)

=q/2m*(L)

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