Show That A length L of wire carries a current i. Show that...

Show That A length $L$ of wire carries a current $i$. Show that if the wire is formed into a circular coil, then the magnitude of the maximum torque in a given magnetic field is developed when the coil has one turn only. Also show that maximum torque has the magnitude $\tau=L^{2} i B / 4 \pi$

Key Concepts

Relationship between Wire Length and Coil Area

For a circular coil made from a fixed wire length, the coil’s circumference directly relates to its radius, and consequently, its area. Specifically, when the wire is formed into a single circular loop, the area can be expressed in terms of the wire length. Maximizing the area (and the resulting magnetic dipole moment) leads to the derived expression for the maximum torque, showing the interplay between geometric constraints and electromagnetic performance.

Coil Geometry and Optimization

Given a fixed length of wire, the geometry of the coil determines the magnetic dipole moment by affecting the loop's area. For a circular coil, the area is maximized when the coil is formed into one turn. This optimization plays an important role because increasing the number of turns while keeping the total wire length constant reduces the area of each turn, ultimately lowering the overall magnetic dipole moment and hence the maximum possible torque.

Torque on a Magnetic Dipole in a Magnetic Field

A magnetic dipole placed in an external magnetic field experiences a torque that tends to align the dipole with the field. This torque is given by the vector cross product of the magnetic dipole moment and the magnetic field, and its magnitude reaches a maximum when the dipole is perpendicular to the field. This concept is crucial for analyzing how current-carrying coils behave in magnetic fields.

Magnetic Dipole Moment

In electromagnetism, a current loop generates a magnetic dipole moment, which is a vector quantity defined as the product of the current and the area of the loop. This dipole moment characterizes the strength and orientation of the magnetic field produced by the current loop, and it is central to predicting the interaction with an external magnetic field.

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