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## Homework Statement

A hypothetical magnetic field existing in a region is given by ##\vec{B}=B_o\vec{e}_r##, where ##\vec{e}_r## denotes the unit vector along the radial direction. A circular loop of radius a, carrying a current ##i##, is placed with its plane parallel to the X-Y plane and the centre at ##(0,0,d)##. Find the magnitude of the magnetic force acting on the loop.

## Homework Equations

Force on a conductor due to a magnetic field : ##\vec{F}_{magnetic}= i\vec{l}##x##\vec{B}##.

## The Attempt at a Solution

A (so far) reliable solution manual (created independent to the referred textbook) plainly explains that as ##\vec{F}_{magnetic}= i\vec{l}##x##\vec{B}## => ##\vec{F}_{magnetic}= i2\pi aB_o(\frac{a}{\sqrt[]{a^2+d^2}})##

Though the final answer is correct, isn't it coincidental and isn't the solution theoretically wrong?

When I've done the problem I think the magnetic field and the length vector are always perpendicular to each other as long as the center of the circular loop is along the z-axis (in the above case) and the magnetic field is radial. So when I resolve the forces I find that component vectors of all the forces acting on the loop cancels out (i.e ##Fcos\theta## around the loop cancels out) whereas the component ## Fsin\theta## adds up and hence the final answer gives : ##\vec{F}_{magnetic}= i2\pi aB_o(\frac{a}{\sqrt[]{a^2+d^2}})##

Am I right? Or am I making a terrible theoretical mistake? Or is the solution manual making a mistake?

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