A wire lying along a y axis from y=0 to y=0.250 m carries a current of 2.00 mA in the negative

step 1 In this question, we have a wire along the y -axis. It has a current in the negative y direction

A wire lying along a y axis from y=0 to y=0.250 m carries a...

A wire lying along a $y$ axis from $y=0$ to $y=0.250 \mathrm{~m}$ carries a current of $2.00 \mathrm{~mA}$ in the negative direction of the axis. The wire fully lies in a nonuniform magnetic field that is given by $\vec{B}=(0.300 \mathrm{~T} / \mathrm{m}) y \hat{\mathrm{i}}+(0.400 \mathrm{~T} / \mathrm{m}) y \mathrm{j} .$ In unit-vector notation, what is the magnetic force on the wire?

Key Concepts

Vector Cross Product in Force Calculations

The vector cross product is a mathematical tool used to calculate the direction and magnitude of the force on a current element in a magnetic field. In the expression dF = I (dl x B), the cross product determines how the relative orientation of the current element and the magnetic field contributes to the overall force. This operation follows the right-hand rule, which is crucial for predicting the direction of the resultant magnetic force.

Magnetic Force on a Current-Carrying Conductor

This concept describes the force experienced by a conductor carrying an electric current when placed in a magnetic field. According to the Lorentz force law for currents, the force on a small segment of the conductor is given by dF = I (dl x B), where I is the current, dl is a differential length element of the wire, and B is the magnetic field. The overall force on the entire wire is obtained by integrating this expression along its length.

Differential Current Element

A differential current element represents an infinitesimally small segment of the wire over which the current flows. In calculations involving nonuniform fields or curved conductors, the element is essential because it allows integration over the geometry of the wire. The element dl has both magnitude and direction and is used in the cross product with the magnetic field to determine the force contribution of that segment.

Nonuniform Magnetic Field

A nonuniform magnetic field is one where the magnitude and/or direction of the field varies with position. Unlike uniform fields, nonuniform fields require that the magnetic force be calculated by integrating the differential forces over the entire spatial extent of the conductor. This variation means that the force on different segments of the wire can differ even if the current is constant, and careful consideration of the field's dependence on position is necessary.

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