A metal alloy is known to have electrical conductivity and...

A metal alloy is known to have electrical conductivity and electron mobility values of $1.2 \times$ $10^{7}(\Omega \cdot \mathrm{m})^{-1}$ and $0.0050 \mathrm{~m}^{2} / \mathrm{V} \cdot \mathrm{s}$, respectively. $\mathrm{A}$ current of $40 \mathrm{~A}$ is passed through a specimen of this alloy that is $35 \mathrm{~mm}$ thick. What magnetic field would need to be imposed to yield a Hall voltage of $-3.5 \times 10^{-7} \mathrm{~V} ?$

Key Concepts

Lorentz Force

The Lorentz force is the force acting on a charged particle moving in the presence of electric and magnetic fields. In the context of the Hall effect, the magnetic component of the Lorentz force deflects moving electrons, leading to a charge imbalance that creates the Hall voltage. Understanding this force is fundamental to analyzing how magnetic fields influence the behavior of charge carriers.

Electron Mobility

Electron mobility quantifies the ease with which electrons can move through a material when subjected to an electric field. It plays a crucial role in determining both the electrical conductivity and the response of the material under external influences, such as magnetic fields. In phenomena like the Hall effect, higher mobility means that the electrons can be more significantly deflected by the Lorentz force.

Current Density and Specimen Geometry

Current density, defined as the current per unit cross-sectional area of a material, is critical for determining the distribution of electrical current through a specimen. The geometry of the specimen, such as its thickness or width, directly affects the current density and subsequently influences the magnitude of the resulting electrical effects, including the Hall voltage. This concept bridges the material properties with the measurable electrical phenomena observed in experiments.

Hall Effect

The Hall effect refers to the generation of a voltage difference (known as the Hall voltage) across a conductor when an electric current flows through it in the presence of a perpendicular magnetic field. This voltage arises because the magnetic field exerts a Lorentz force on the moving charge carriers, causing them to deflect to one side of the conductor, thus creating a potential difference transverse to both the current and the magnetic field.

Electrical Conductivity

Electrical conductivity is a measure of how readily a material allows the flow of electric charge. It is determined by the density of charge carriers within the material and how effectively these charges can move. A high conductivity indicates that electrons can flow easily, which is essential for understanding the distribution of current in an electrical system and its response to applied fields.

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