Prove that the relation τ=N i A B sinθholds not only for the rectangular loop of Fig. 28-19 but

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Prove that the relation τ=N i A B sinθholds not only for the...

Prove that the relation $\tau=N i A B \sin \theta$ holds not only for the rectangular loop of Fig. 28-19 but also for a closed loop of any shape. (Hint: Replace the loop of arbitrary shape with an assembly of adjacent long, thin, approximately rectangular loops that are nearly equivalent to the loop of arbitrary shape as far as the distribution of current is concerned.)

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that characterizes the strength and orientation of a current loop’s magnetic properties. It is defined as the product of the current and the area enclosed by the loop, with its direction given by the right-hand rule. This concept is central to determining the magnetic behavior of loops regardless of their shape, as it simplifies the description of their interaction with external magnetic fields.

Torque on a Magnetic Dipole

When a magnetic dipole is placed in a magnetic field, it experiences a torque that tends to align the dipole moment with the field. This torque is given by the cross product of the magnetic dipole moment and the magnetic field, leading to the relationship ? = m × B, where m = iA. This expression shows that the torque depends on the magnetic moment’s magnitude, the magnetic field strength, the sine of the angle between them, and the number of loops if multiple turns are present.

Superposition Principle

The superposition principle states that the net effect produced by multiple individual contributions is the sum of the effects produced by each contribution independently. In the context of the problem, an arbitrary current loop can be seen as an assembly of many small, nearly rectangular loops whose individual magnetic moments add vectorially to give the total magnetic moment, ensuring that the overall torque relationship remains valid regardless of the loop’s shape.

Decomposition of Complex Loop Shapes

This concept involves breaking down a complex or arbitrarily shaped current loop into simpler, more manageable segments—such as long, thin rectangular pieces—whose magnetic properties are well understood. By ensuring that the distribution of current is maintained across these segments, one can prove that the overall magnetic dipole moment and the resulting torque are the same as those computed for a simple rectangular loop, thereby generalizing the formula to loops of any shape.

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