A long-rigid wire lies along the x -axis and carries a current of 2.5 A in the positive x -direc

step 1 Force is given by integration of the differential forces which is I times d l cross B equals cur

A long-rigid wire lies along the x -axis and carries a...

A long-rigid wire lies along the $x$ -axis and carries a current of $2.5 \mathrm{A}$ in the positive $x$ -direction. Around the wire is the magnetic field $\overrightarrow{\mathbf{B}}=2.0 \hat{\mathbf{i}}+5.0 x^{2} \hat{\mathbf{j}}, \quad$ with $x$ in meters and $B$ in millitesla. Calculate the magnetic force on the segment of wire between $x=2.0 \mathrm{m}$ and $x=4.0 \mathrm{m}$.

Key Concepts

Handling Spatially Varying Magnetic Fields

In many practical situations, the magnetic field is not uniform; it can vary with position. Recognizing and correctly incorporating these variations into calculations is critical. This involves expressing the magnetic field as a function of spatial coordinates and integrating, ensuring that every element of the current-carrying wire interacts with the appropriate local field, which ultimately affects the net force experienced by the wire.

Line Integration in a Variable Magnetic Field

When the magnetic field varies with position, calculating the total force on the conductor involves integrating the differential force along the segment of the wire. Line integration sums up the contributions from each infinitesimal section of the wire, accounting for the spatial variability of the magnetic field to determine the net force.

Lorentz Force on a Current-Carrying Conductor

This concept describes the force experienced by a current-carrying conductor placed in a magnetic field. The force is given by dF = I (dL × B), where I is the current, dL is a differential element of the conductor’s length, and B is the magnetic field. This law is a fundamental principle in electromagnetism and is essential for understanding how currents interact with magnetic fields.

Vector Cross Product

The vector cross product is a mathematical operation used to determine the direction and magnitude of the magnetic force on the current element. In the context of the Lorentz force, it ensures that the force is perpendicular to both the direction of the current and the magnetic field. A correct application of the cross product is crucial for obtaining the right orientation and magnitude of the force in three-dimensional space.

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