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An insulating rod of length l carries a charge q distributed uniformly on it.The rod is pivoted at an end and is rotated at a frequency f about a fixed perpendicular axis.The magnetic moment of the system is?
let the charge density on rod be λ = q/l consider a small element of rod dx at a distance x from the fixed end charge on it dq = λ dx when the rod rotates the current due to this element , di = f dq = f λ dx magnetic moment of this current element dM = di * area = f λ dx * ∏ x^2 total magnetic moment M = ∫ dM = = ∫0 l f λ dx * ∏ x^2 = fλ∏ ∫0 l ( x^2 ) dx = fλ∏/3 * x^3 ]0 l = fλ∏ l^3 /3 λ l = q , so M = f∏q l^2 /3
let the charge density on rod be λ = q/l
consider a small element of rod dx at a distance x from the fixed end
charge on it dq = λ dx
when the rod rotates the current due to this element , di = f dq
= f λ dx
magnetic moment of this current element dM = di * area
= f λ dx * ∏ x^2
total magnetic moment M = ∫ dM =
= ∫0 l f λ dx * ∏ x^2
= fλ∏ ∫0 l ( x^2 ) dx
= fλ∏/3 * x^3 ]0 l
= fλ∏ l^3 /3
λ l = q , so
M = f∏q l^2 /3
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