Charge and Current

Introduction

This post is about electromagnetism and the continuity equation.

While studying electromagnetism from Giancoli’s book I encountered the concepts of charge density $\rho$ and current density $\vec{j}$. The charge density is a signed quantity that gives the infinitesimal charge at $x$ in time $t$, while the current density is a vector that denotes the infinitesimal velocity of that infinitesimal charge, multiplied by the charge itself. (a kind of momentum.) Suddenly I wondered why is it that $\vec{j}$ cannot be derived by $\rho$?

This conviction was based on the discrete situation, where a single charge $q$ is given by its position $x(t)$, and then $\vec j$ is equal to $q x’(t)$.

However, the continuous case is not as apparent! Naive attempts such as $\vec j = \partial_t \rho$ or $\vec j = \partial_t\nabla\rho$ were incorrect. I tested them on heat dissipation, i.e. the heat kernel $(4\pi k t)^{-1/2}\exp(-\frac{x^2}{4kt})$ and that’s how I knew that I didn’t have the right concept.

Frustrated, I searched online for answers and encountered the relationship between charge and density: the continuity equation! In fact, this equation exactly states the relationship between these two quantities, and captures the intuition that current density must be derivable from charge density.

The equation is $\partial_t \rho = -\operatorname{div}\vec{F}$. The left hand side is the rate of change of charge, while divergence is a measure of outgoingness of the field $\vec{F}$. For example, if charge is lessening at $x$, and so $\partial_t\rho$ is negative, we expect the current density to be outgoing. The current density $\vec{j}$ satisfies this equation. (it is implied by Maxwell’s equations.)

Some more thoughts

For me, this is one of the few interesting differential equations that I have come across, as I have now related it to my intuition and my mistaken naive attempts to calculate current from charge.