A hypothetical magnetic field existing in a region is given by $\vec{B} = B_{0} {\vec{e}}_{r},$ 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.

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Solution

Given:
A hypothetical magnetic field existing in a region, $\vec{B} = B_{0} {\vec{e}}_{r}$
where ${\vec{e}}_{r}$ denotes the unit vector along the radial direction.
A circular loop of radius a
So, the length of the loop, l = 2πa
Electric current through loop = i
As per the question, the loop is placed with its plane parallel to the X−Y plane and its centre is at (0, 0, d).

Here, angle between the length of the loop and the magnetic field is θ.
Magnetic force is given by
$\vec{F} = i \vec{l} \times \vec{B} \vec{F} = i (2 π a \times \vec{B}) \vec{F} = i 2 π a B sinθ = \frac{i 2 π a B_{0} a}{\sqrt{a^{2} + d^{2}}} = \frac{i 2 π B_{0} a^{2}}{\sqrt{a^{2} + d^{2}}}$

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