A current loop $ABCD$ is held fixed on the plane of the paper as shown in the figure. The arcs $BC$ (radius $= b$) and $DA$ (radius $=a$) of the loop are joined by two straight wires $AB$ and $CD$. A steady current $I$ is flowing in the loop. Angle made by $AB$ and $CD$ at the origin $O$ is ${30^ \circ }.$ Another straight thin wire steady current ${I_1}$ flowing out of the plane of the paper is kept at the origin. Due to the presence of the current ${I_1}$ at the origin:

Question

A current loop $ABCD$ is held fixed on the plane of the paper as shown in the figure. The arcs $BC$ (radius $= b$) and $DA$ (radius $=a$) of the loop are joined by two straight wires $AB$ and $CD$. A steady current $I$ is flowing in the loop. Angle made by $AB$ and $CD$ at the origin $O$ is ${30^ \circ }.$ Another straight thin wire steady current ${I_1}$ flowing out of the plane of the paper is kept at the origin.

Due to the presence of the current ${I_1}$ at the origin:

The forces on $AD$ are $BC$ are zero.

The magnitude of the net force on the loop is given by $${{{I_1}I} \over {4\pi }}{\mu _0}\left[ {2\left( {b - a} \right) + {\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 3$}}\left( {a + b} \right)} \right].$$

The magnitude of the net force on the loop is given by ${{{\mu _0}I{I_1}} \over {24ab}}\left( {b - a} \right).$

The forces on $AB$ and $DC$ are zero.

Solution

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