Electric Current

If voltage is applied to a conductor, electrons move from the negative to the positive pole as shown in the following figure. Each circle represents a single electron and the arrows symbolize the direction of motion. The current direction presented in this figure is the physical current direction. However, in the past, it was believed that electric current flows from the positive to the negative pole. Therefore, the conventionally agreed direction, also called technical current direction, points from plus to minus. We want to quantify the term now and find the following definition for the electric current. Here, $Q$ is the full amount of charge, i.e. all electrons moving inside the wire. Since the current in a wire is created by moving electrons, this formula can be rewritten as $$I = \frac{ne}{t}$$ for a constant current. Here, $n$ stands for the number of electrons and $e$ for the elementary charge. Sometimes we are not interested in the total current, but rather in the current per unit area. Another important quantity is therefore the current density $\vec{j}$ which is a vector quantity in general. We want to define it via the dot product of the vectors $\vec{j}$ and $\vec{A}$. Although this quantity is called density, it is important to remember that it refers to the amount of charge per area and not per volume. The current density can therefore be written as $$j = \frac{\mathrm{d}I}{\mathrm{d}A}$$ Substituting $I$ with $\mathrm{d}Q/\mathrm{d}t$ gives $$j = \frac{\mathrm{d}^2Q}{\mathrm{d}A\,\mathrm{d}t}$$ In the next step, we replace $\mathrm{d}Q$ with the product $\varrho \mathrm{d}V$ where $\varrho$ is the charge density that is assumed to be constant. This leads to the following expression of the charge density: $$j = \varrho \frac{\mathrm{d}V}{\mathrm{d}t\,\mathrm{d}A}$$ We can now rewrite $\mathrm{d}V/\mathrm{d}A$ with $\mathrm{d}s$: $$j = \frac{\mathrm{d}s}{\mathrm{d}t}$$ Replacing $\mathrm{d}s/\mathrm{d}t$ with the speed $v$ of the charge finally results in $$j = \varrho v$$ This is a very important formula to compute the current density if the density and speed of the charges are known. We can of course rewrite this statement in vector form to generalize it further.