In Fig. 28-8, show that the ratio of the Hall electric field...

In Fig. $28-8,$ show that the ratio of the Hall electric field magnitude $E$ to the magnitude $E_{c}$ of the electric field responsible for moving charge (the current) along the length of the strip is $$ \frac{E}{E_{C}}=\frac{B}{n e \rho} $$ where $\rho$ is the resistivity of the material and $n$ is the number density of the charge carriers. (b) Compute this ratio numerically for Problem 13.

In Fig. $28-8,$ show that the ratio of the Hall electric field magnitude $E$ to the magnitude $E_{c}$ of the electric field responsible for moving charge (the current) along the length of the strip is
$$
\frac{E}{E_{C}}=\frac{B}{n e \rho}
$$
where $\rho$ is the resistivity of the material and $n$ is the number density of the charge carriers. (b) Compute this ratio numerically for Problem 13.

Key Concepts

Hall Effect

The Hall effect is the generation of a transverse voltage (the Hall voltage) across a conductor when it carries an electric current in the presence of a perpendicular magnetic field. This phenomenon arises because the magnetic field exerts a Lorentz force on the moving charge carriers, deflecting them to one side of the conductor, which creates a potential difference that can be measured and used to determine properties like the charge carrier density.

Electric Field

An electric field is a vector field that represents the force per unit charge exerted on charged particles. In the context of current flow through a conductor, the electric field is responsible for driving the movement of charge carriers along the material. The applied or driving electric field is crucial in establishing the current, while in the Hall effect, a secondary electric field develops perpendicular to both the current and the magnetic field.

Resistivity

Resistivity is a fundamental material property that quantifies how strongly a material opposes the flow of electric current. It links the electric field within a material to the current density via Ohm's law. In problems involving the Hall effect, resistivity helps determine how the applied electric field relates to the current and, consequently, how the induced Hall field competes with the driving field.

Charge Carrier Density

Charge carrier density refers to the number of mobile charge carriers (such as electrons or holes) per unit volume in a material. This parameter is essential in determining the electrical conductivity of the material and directly affects the magnitude of the Hall voltage, since the separation of charge under the influence of a magnetic field depends on how many carriers are available to be deflected.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in the presence of electric and magnetic fields. Specifically, the magnetic component of this force acts perpendicular to both the velocity of the charge and the magnetic field direction, which is the driving mechanism behind the Hall effect. The balance between the Lorentz force and the electric force due to the Hall field leads to the establishment of a steady state in which the transverse voltage can be observed.

Magnetic Field

A magnetic field is a vector field that influences the motion of moving charges and magnetic dipoles. In the Hall effect, the magnetic field plays a pivotal role by exerting a force on the moving charge carriers (via the Lorentz force), causing their deflection and resulting in the buildup of a measurable transverse electric field. The strength of the magnetic field directly influences the magnitude of the Hall voltage.

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