Question

What is j in Maxwell equations?

Answer 1

In Maxwell's equations, "j" represents the current density.

More specifically, current density refers to the amount of electric current flowing per unit area of a material. It’s a vector quantity, meaning it has both a magnitude and a direction. The symbol j is commonly used in the equations, and it is typically measured in units of amperes per square meter (A/m²).

Maxwell's Equations Involving "j":

1. Gauss's Law for Electricity (Electric fields due to charge):
   

   \[
   \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}
   \]
   (Here, \(\rho\) is the charge density, not directly related to "j", but the current density affects electric fields via charge distribution.)

1. Ampère's Law with Maxwell's Addition (Magnetic fields created by currents and changing electric fields):
   

   \[
   \nabla \times \mathbf{B} = \mu_0 \mathbf{j} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}
   \]
   In this equation, j is the current density. It describes the flow of electric current that generates magnetic fields. The term \(\mu_0\) is the permeability of free space, and \(\epsilon_0\) is the permittivity of free space. The term \(\frac{\partial \mathbf{E}}{\partial t}\) represents the changing electric field, which can also induce magnetic fields.

Key Points About "j" (Current Density):

1. j is a vector, meaning it has both magnitude (how much current) and direction (where the current is flowing).
   

1. It is crucial in understanding the relationship between electric currents and magnetic fields (via Ampère's Law).
   

1. Current density is related to the electric field and the conductivity of the material in Ohm’s Law:
   

  \[
  \mathbf{j} = \sigma \mathbf{E}
  \]
  where σ is the electrical conductivity of the material.

In summary, j in Maxwell's equations is the current density, and it plays a key role in linking electric currents with magnetic fields.