The conduction current density in a liquid metal is $$ \mathbf{J}_{f}=\sigma(\mathbf{E}+\mathbf{v} \times \mathbf{B}) $$ where \(\sigma\) is the conductivity, \(\mathbf{E}\) is the electric field intensity, \(\mathbf{v}\) is the velocity of the fluid, and \(\mathbf{B}\) is the magnetic induction, all quantities measured in the frame of reference of the laboratory. Show that the magnetic force per unit volume of fluid is $$ \sigma(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \times \mathbf{B} $$ For example, in crossed electric and magnetic fields, a conducting fluid is pushed in the direction of \(\mathbf{E} \times \mathbf{B}\). The field \(\mathbf{v} \times \mathbf{B}\) then opposes \(\mathbf{E}\). This is the principle of operation of electromagnetic pumps.

The Lorentz force is a fundamental concept in electromagnetism, describing the force exerted on a charged particle by electric and magnetic fields. It is mathematically represented as \( \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \) where:

• \( q \) is the electric charge of the particle,
• \( \mathbf{E} \) is the electric field intensity,
• \( \mathbf{v} \) is the velocity of the particle,
• \( \mathbf{B} \) is the magnetic field (also referred to as magnetic induction).

When considering the cumulative effect on an entire volume of charged particles, such as those in a fluid or a metal, we deal with the Lorentz force per unit volume. This is obtained by replacing the charge \( q \) with the current density vector \( \mathbf{J} \), leading to \( \mathbf{f} = \mathbf{J} \times \mathbf{B} \) which gives us a measure of the force density acting on the substance due to the electric and magnetic fields.

Understanding this fundamental interaction is crucial for analyzing the behavior of charged particles in various fields, including electrical engineering and plasma physics.

Conductivity, denoted by the symbol \( \sigma \), is a measure of a material's ability to conduct electric current. It is the reciprocal of resistivity, representing how easily electrons can flow through a material. High conductivity implies that the material allows electric charges to move freely through it, making it a good conductor. Conversely, low conductivity indicates a poor conductor, which impedes electron flow.

In the context of electromagnetism and the Lorentz force, conductivity plays a role in determining the current density within a material, as seen in the formula \( \mathbf{J}_f = \sigma(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \). Here, the current density vector \( \mathbf{J}_f \) is directly proportional to the conductivity; higher conductivity leads to a higher current density for the same electric and magnetic field magnitudes. Understanding the conductivity of a material is essential when designing electrical systems and components, such as wires, circuits, or even the aforementioned electromagnetic pumps.

Electric field intensity, usually denoted as \( \mathbf{E} \), is a fundamental concept that describes the electric force per unit charge exerted at any point in space. It is a vector quantity, having both magnitude and direction, and represents how strongly an electric field will influence charges within it. The field intensity is determined by the distribution of electric charges and their distances from the point in question.

In the formula for the Lorentz force and the current density vector, \( \mathbf{E} \) represents the 'electric part' of the electromagnetic interaction. Any charge placed in an electric field \( \mathbf{E} \) will experience a force \( q\mathbf{E} \) along the field's direction. The role of the electric field intensity is crucial in many applications, from determining the path of electrons in a cathode ray tube to calculating the forces on charges in electrical machinery.

The current density vector, denoted as \( \mathbf{J} \), is a vector quantity that describes the flow of electric charge in terms of its magnitude and direction per unit area. For a given cross-sectional area through which charge flows, the current density measures how dense the flow of charge is at each point. It is represented mathematically as \( \mathbf{J} = \sigma(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \) for a moving conductor or charge carrier in a fluid, where \( \sigma \) is the conductivity, \( \mathbf{E} \) is the electric field intensity, \( \mathbf{v} \) is the velocity, and \( \mathbf{B} \) is the magnetic field.

In electromagnetism, the concept of current density is used to understand and predict the behavior of electric currents in various mediums, ranging from solid metals to moving liquids. It is also intimately tied to the concept of the Lorentz force, as it forms a part of the expression calculating the force per unit volume in a substance subjected to electric and magnetic fields.