Electric charge

Electric charge ($q$) is a property of matter, measured in coulombs ($C$). We assign positive charges to protons, neutral charges to neutrons, and negative charges to electrons. Crucial properties are that charge is quantised (discrete multiples of the elementary charge), and conserved (does not change in an isolated system).

Like charges repel, opposite charges attract; this force is massive (about $10^{36}$ times stronger than gravity), so any stable matter is electrically neutral. Stable mixtures are formed by balancing out these repulsive and attractive forces.

We define the elementary charge as:

$$e=1.602\times 10^{-19}C$$

All particles have an integer multiple of $e/3$, the smallest possible charge of a quark. Electrons have a charge of $-e$, protons have a charge of $e$.

Representations

Since charge within particles are concentrated within a super small volume, we can approximate a particle as a small point charge. This allows us to assign spatial coordinates and a position vector $\vec{R}$ to the point.

Similarly, while charge distributions are discrete at an atomic scale, we approximate them as continuous and uniform in our analysis. We can find the enclosed charge by integrating the density with respect to the differential element on the denominator.

The volume charge density is given by (in $C\cdot m^{-3}$), i.e., an electron:

$${\rho }_v=\frac{dq}{dv}$$

The surface charge density is given by (in $C\cdot m^{-2}$), i.e., a capacitor’s plates:

$$\sigma ={\rho }_s=\frac{dq}{ds}$$

The line charge density is given by given by (in $C\cdot m^{-1}$), i.e., in cables or power lines:

$$\lambda ={\rho }_l=\frac{dq}{dl}$$

Addendums

We can relate to current:

$$q(t)={\int }_{\infty }^{t}i(x)dx$$

The charge of the carriers in a length of wire is:

$$q=(nAL)e$$

In electronics, we tend to work with smaller units: $\mu C$ or $nC$.

Gauss’ law finds the charge enclosed by a closed surface (a Gaussian surface).