In Fig. 29-46, two concentric circular loops of wire...

In Fig. $29-46,$ two concentric circular loops of wire carrying current in the same direction lie in the same plane. Loop 1 has radius $1.50 \mathrm{~cm}$ and carries $4.00 \mathrm{~mA}$. Loop 2 has radius $2.50 \mathrm{~cm}$ and carries $6.00 \mathrm{~mA}$. Loop 2 is to be rotated about a diameter while the net magnetic field $\vec{B}$ set up by the two loops at their common center is measured. Through what angle must loop 2 be rotated so that the magnitude of that net field is $100 \mathrm{nT} ?$

Key Concepts

Magnetic Field of a Circular Current Loop

A circular loop carrying an electric current produces a magnetic field along its axis. Using the Biot–Savart law, one can derive that the magnetic field at the center of the loop is proportional to the current and inversely proportional to the loop’s radius. This relation forms the basis for calculating the induced magnetic field in many electromagnetic applications.

Superposition Principle in Magnetism

In electromagnetism, the superposition principle states that the net magnetic field at a point due to multiple sources is the vector sum of the individual fields produced by each source independently. This concept is crucial when dealing with systems involving several current-carrying loops or coils, as it allows for the calculation of the overall field by combining the separate contributions.

Effect of Orientation on Magnetic Field Contribution

When a current-carrying loop is rotated relative to a given direction, the effective contribution of its magnetic field in that direction is reduced. The magnitude of the component of the field along a particular axis is given by the original magnitude multiplied by the cosine of the angle between the field direction and that axis. This trigonometric dependence is central to understanding how the field changes with rotation.

Vector Projection and Trigonometric Relationships in Field Analysis

Magnetic fields are vector quantities, and understanding their direction and magnitude involves breaking them into components. The cosine function arises naturally when projecting a vector onto an axis, which is essential when analyzing how rotating one of the sources affects the net magnetic field. This approach is widely used in various areas of physics to resolve the effects of vector rotation.

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