A length l of wire carries a steady current i. It is bent...

A length $l$ of wire carries a steady current $i$. It is bent first to form a circular plane coil of one turn. The same length is now bent more sharply to give three loops of smaller radius. The magnetic field at the centre caused by the same current is (a) one-third of its value (b) unaltered (c) three times of its initial value (d) nine times of its initial value

A length $l$ of wire carries a steady current $i$. It is bent first to form a circular plane coil of one turn. The same length is now bent more sharply to give three loops of smaller radius. The magnetic field at the centre caused by the same current is
(a) one-third of its value
(b) unaltered
(c) three times of its initial value
(d) nine times of its initial value

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Key Concepts

Magnetic Field from a Circular Current Loop

A circular loop of current produces a magnetic field at its center that depends on the current, the permeability of free space, and the radius of the loop. This fundamental concept in electromagnetism is used to calculate the field strength produced by a ring carrying a steady current and forms the basis for understanding magnetic fields in various coil configurations.

Wire Length and Coil Geometry

When a fixed length of wire is used to form a circular coil, the circumference of the loop is constant, which directly determines its radius. Altering the number of loops while keeping the wire length unchanged changes the radius of each individual loop. This relationship between the wire length and the geometry of the coil is critical for analyzing how the coil’s dimensions affect its magnetic properties.

Combined Effect of Multiple Turns

In coils with several turns, the total magnetic field at the center is the sum of the fields produced by each turn. This principle of superposition means that if the individual contributions are similar and in phase, the overall field strength scales with the number of turns. This concept is essential in the design of electromagnets and solenoids, where increasing the number of turns enhances the magnetic field.

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