A wire lying along an x axis from x=0 to x=1.00 m carries a current of 3.00 A in the positive

step 1 In this question, we have a wire that is along the x -axis. Here is x equals to 0, here is x equ

A wire lying along an x axis from x=0 to x=1.00 m carries a...

A wire lying along an $x$ axis from $x=0$ to $x=1.00 \mathrm{~m}$ carries a current of $3.00 \mathrm{~A}$ in the positive $x$ direction. The wire is immersed in a nonuniform magnetic field that is given by $\vec{B}=$ $\left.\left(4.00 \mathrm{~T} / \mathrm{m}^{2}\right) x^{2} \hat{\mathrm{i}}-\left(0.600 \mathrm{~T} / \mathrm{m}^{2}\right) x^{2}\right] .$ In unit-vector notation, what is the magnetic force on the wire?

Key Concepts

Magnetic Force on a Current?Carrying Conductor

This concept involves the force experienced by a current-carrying wire when placed in a magnetic field. The force is given by the equation F = I (L × B), where I is the current, L is a vector representing the length and direction of the current element, and B is the magnetic field. This relationship is foundational in electromagnetism and explains how moving charges interact with magnetic fields.

Differential Current Elements and Integration

When the magnetic field is nonuniform, the force on different segments of the wire may differ. Therefore, the wire is divided into infinitesimal segments (differential current elements), and the force on each element (dF = I (dl × B)) is calculated. The total force is then determined by integrating these differential forces along the length of the wire.

Cross Product in the Lorentz Force Law

The magnetic force involves the cross product between the current element and the magnetic field. The cross product operation inherently accounts for the perpendicular nature of the magnetic force relative to both the current direction and the magnetic field direction. Understanding vector cross products is essential for determining the direction and magnitude of the magnetic force in three dimensions.

Non?Uniform Magnetic Field

A nonuniform magnetic field is one whose magnitude and/or direction varies with position. In problems of this type, the magnetic field is expressed as a function of position, meaning that the force on different segments of the conductor must be calculated taking into account these variations. This requires setting up and solving an integral over the length of the wire to find the total force.

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