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(D) The torque $\vec{	au}$ acting on a current-carrying loop in an external magnetic field $\vec{B}$ is given by the formula $\vec{	au} = \vec{m}

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External magnetic field

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(D) The torque $\vec{	au}$ acting on a current-carrying loop in an external magnetic field $\vec{B}$ is given by the formula $\vec{	au} = \vec{m} 	imes \vec{B}$,where $\vec{m}$ is the magnetic dipole moment.The magnetic moment is defined as $\vec{m} = I \vec{A}$,where $I$ is the current and $\vec{A}$ is the area vector of the loop.Substituting this into the torque equation,we get $\vec{	au} = I(\vec{A} 	imes \vec{B})$.From this expression,it is clear that the torque depends on the current $(I)$,the area vector $(\vec{A})$,and the external magnetic field $(\vec{B})$.While the area vector depends on the shape and size of the loop,for a fixed length of wire,the maximum area is enclosed when the shape is circular. However,the torque itself is directly proportional to the area vector $\vec{A}$.Since the torque depends on $I$,$\vec{A}$,and $\vec{B}$,it does not depend on the specific shape of the loop as long as the area remains constant. However,the question asks what it does not depend on among the given options. Since torque depends on all listed factors (Shape affects $\vec{A}$,$\vec{A}$ is area,$I$ is current,$\vec{B}$ is field),and the options provided in the original question structure are slightly ambiguous,the correct interpretation is that torque depends on all these factors. Therefore,the answer is 'None' of the above.

$A$ loop is formed by a wire of constant length carrying a current and placed in an external magnetic field. The torque acting on it does not depend upon:

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