A wire carries a steady current of 2.40 A. A straight...

A wire carries a steady current of 2.40 A. A straight section of the wire is 0.750 $\mathrm{m}$ long and lies along the $x$ axis within a uniform magnetic field, $\overrightarrow{\mathbf{B}}=1.60 \hat{\mathbf{k}} \mathrm{T}$ . If the current is in the $+x$ direction, what is the magnetic force on the section of wire?

Key Concepts

Uniform Magnetic Field

A uniform magnetic field maintains a constant magnitude and direction throughout a given region. This simplification allows for straightforward calculations of the magnetic force on a current-carrying wire since the field does not vary along the length of the wire. Such conditions are often assumed in textbook problems to elucidate the fundamental principles of magnetic interactions.

Magnetic Force on a Current-Carrying Conductor

This concept describes how a current-carrying wire experiences a force when placed in a magnetic field. The force is calculated using the formula F = I (L x B), where I represents the current, L is the vector representing the length and direction of the wire, and B is the magnetic field vector. This is a fundamental principle in electromagnetism, especially in the design and operation of devices like electric motors and generators.

Lorentz Force Law

The Lorentz force law governs the behavior of charged particles and current-carrying conductors in electromagnetic fields. For a current element in a magnetic field, the force is given by F = I (L x B). This law provides a quantitative description of how magnetic and electric fields interact with moving charges, forming the basis for understanding electromagnetic forces.

Vector Cross Product

The vector cross product is a mathematical operation used to determine a vector that is perpendicular to two original vectors. In the context of a current-carrying wire in a magnetic field, the cross product L x B determines both the magnitude and direction of the force acting on the wire. The magnitude is given by I L B sin(?), where ? is the angle between L and B.

Right-Hand Rule

The right-hand rule is a mnemonic that helps determine the direction of the vector resulting from a cross product. When applying this rule to the formula F = I (L x B), the fingers of the right hand point in the direction of the current (L), and when they curl toward the magnetic field (B), the extended thumb indicates the direction of the magnetic force. This is crucial for correctly predicting the orientation of forces in electromagnetic applications.

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