A horizontal wire of length 0.53 m, carrying a current of 7.5 A, is placed in a uniform external

step 1 In this problem, we're given two situations of a current carrying wire being present around a ma

A horizontal wire of length 0.53 m, carrying a current of...

A horizontal wire of length $0.53 \mathrm{m},$ carrying a current of $7.5 \mathrm{A},$ is placed in a uniform external magnetic field. When the wire is horizontal, it experiences no magnetic force. When the wire is tilted upward at an angle of $19^{\circ},$ it experiences a magnetic force of $4.4 \times 10^{-3} \mathrm{N}$ . Determine the magnitude of the external magnetic field.

Key Concepts

Magnetic Force on a Current-Carrying Conductor

This concept involves the force experienced by a conductor carrying an electric current when placed within a magnetic field. The force is given by the equation F = I L B sin(?), where I is the current, L is the length of the conductor, B is the magnetic field strength, and ? is the angle between the conductor’s length and the magnetic field direction. This relation highlights how both the magnitude of the current and its orientation relative to the magnetic field determine the force experienced.

Vector Cross Product and Angle Dependence

This concept is based on the mathematical cross product which shows that the force on the conductor depends on the sine of the angle between the current direction and the magnetic field. When the conductor is aligned parallel to the magnetic field (? = 0°), sin(?) is zero and thus no force is experienced. When the conductor is tilted relative to the field, sin(?) becomes non-zero, resulting in a measurable force. This explains how changing the orientation of the conductor affects the magnitude of the magnetic force.

Uniform Magnetic Field

Understanding a uniform magnetic field is crucial as it implies that the magnetic field has the same magnitude and direction at every point within the region of interest. This uniformity simplifies the calculation of the magnetic force since the field strength B can be treated as constant along the entire length of the conductor. It provides a clear context for applying the force equation without having to account for variations in the magnetic field.

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