notes/education/physics/PHYS2220/Electric Charge.md

Electric Charge

• Charges come in two varieties, positive and negative.
• Net charge is the algebraic sum of an object's charges
• Protons and electrons have the same magnitude of charge (designated 1e; a unit, not Euler's number)
  • The SI Unit of charge is the Coulomb (abbreviated C)
  • The smallest discrete quantity of charge is \frac{1}{3}e.
• In an isolated system, the net charge will always remain constant.

Coulomb's Law

• Two charges will exert a force on each other along the line joining them.
  • The magnitude of this force is proportional to the product of the charges and inversely proportional to the to the \sqrt{dist}.
  • The equation to determine the force between two charges is as follows:
  vec{F}_{12} = \vec{r}k\frac{q_1q_2}{r^2} $$

    - $\vec{r}$ is a unit vector pointing from charge 1 to charge 2
    - $k$ is Coulomb's constant, or $8.99 * 10^9 \frac{Nm^2}{C^2}$ 
    - $q_1$ and $q_2$ are the charges
    - $r$ is the distance between those charges
    - The resulting force will push away if $q_1q_2$ is *positive*, and attract if $q_1q_2$ is negative. This is where the rule "opposites attract, like repels" comes from
  

• Coulomb's law only holds exactly true for point charges i.e a proton or electron.

The Superposition Principle

The superposition principle states that:

The net force acting on a point charge is equal to the sum of all individual forces.

This means that to find the net force acting on a single charge, you add up all of the individual forces acting on that charge.

The Electric Dipole

An electric dipole consists of two point charges of equal magnitude but opposite sign. Many molecules behave like dipoles.

• Electric dipole moment (p) is defined as the product of the charge q and the separation d between the two charges making up the dipole. p = qd
• The dipole field at large distances decreases as the inverse cube of the distance. This is because the dipole has zero net charge.
• In an electric field, a dipole experiences a torque that aligns it with the field.

Continuous Charge Distributions

It's largely impossible to sum the electric field from every particle in a piece of matter, so the approximation is made that the charge is spread continuously over the distribution.

• The number of dimensions involved changes the unit and terminology used:
  • If the charge distribution extends throughout a 3d volume, we describe it in terms of the volume charge density \rho, with units of \frac{C}{m^3}.
  • For charge distributions spread over surfaces, we use surface charge density \sigma (\frac{C}{m^2}).
  • For charge distributions spread over lines, we use line charge density \lambda (\frac{C}{m}).
• To find the point charge, we can use this formula:

\vec{E} = \int d \vec{E} = \int \frac{kdq}{r^2}\hat{r}