A long, rigid conductor, lying along an x axis, carries a current of 7.0 A in the negative x di

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A long, rigid conductor, lying along an x axis, carries a...

A long, rigid conductor, lying along an $x$ axis, carries a current of $7.0 \mathrm{~A}$ in the negative $x$ direction. A magnetic field $\vec{B}$ is present, given by $\vec{B}=3.0 \mathrm{i}+8.0 x^{2} \hat{\mathrm{j}}$, with $x$ in meters and $\vec{B}$ in milliteslas. Find, in unit-vector notation, the force on the $2.0 \mathrm{~m}$ segment of the conductor that lies between $x=1.0 \mathrm{~m}$ and $x=3.0 \mathrm{~m}$.

Key Concepts

Unit Vector Notation

Unit vector notation is used to express directions in space with standard basis vectors, such as i, j, and k. It provides a concise method for representing vectors in problems involving physical quantities like magnetic field and force, allowing for a clear description of each vector’s components along the coordinate axes.

Magnetic Force on a Current-Carrying Conductor

This concept involves the force that acts on a conductor carrying an electric current when it is placed in a magnetic field. The force is given by F = I (L × B), where I is the current, L is a vector representing the length and direction of the conductor, and B is the magnetic field. This relationship is central in electromagnetism and is an example of how electric currents interact with magnetic fields.

Lorentz Force Law (Differential Form)

When the magnetic field is not uniform along the length of the conductor, the force on an infinitesimal segment of the conductor must be calculated using the differential form dF = I (dl × B). This approach allows one to account for variations in the magnetic field by considering small segments, each experiencing a slightly different magnetic force, and then integrating them over the desired length.

Integration over a Conductor in a Variable Magnetic Field

If the magnetic field varies with position, the overall force on the conductor segment is determined by integrating the differential magnetic force along the path of the conductor. This process, which involves summing contributions of infinitesimal force elements, is essential when dealing with spatially dependent fields.

Cross Product in Vector Calculus

The cross product is a mathematical operation that determines a vector perpendicular to two given vectors. In the context of determining magnetic force, the cross product between the current element (or length vector) and the magnetic field vector establishes both the magnitude and the direction of the force, following the right-hand rule.

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