Vector Calculus Recap
A field is a physical quantity that is defined at each point in time and space. It could be a scalar, vector, or tensor. Temperatures are an example of a scalar field and wind an example of vector fields.
Gradient (vector showing direction of steepest ascent), divergence (scalar), and curl (vector showing how much rotation) are vector field operations.
Charge
Charge has some fundamental properties:
- There are 2 types of charge.
- Charges are quantized.
- Charges are conserved.
Coulomb’s Law
We can write Coulomb’s Law:
$$\mathbf{F}_1 = k\frac{q_1q_2}{r_{12}^2}\hat{\mathbf{r}}_{12}.$$
Alternatively, we could substitute \(\frac{1}{4\pi\epsilon_0}\) for \(k\). By comparing \(Gm^2\) and \(kq^2\) for a proton, we can see that for two protons, the electric force is much stronger than the gravitational force.
Rearranging Coulomb’s Law for three total points gives us
$$\mathbf{F}_1 = k\frac{q_1q_3}{r_{13}^2}\hat{\mathbf{r}}_{13} + k\frac{q_1q_2}{r_{12}^2}\hat{\mathbf{r}}_{12},$$
which implies that we can calculate separate forces on \(q_1\) and add them together (remember we are adding vectors). This is known as superposition, which holds true for classical mechanics but not necessarily quantum mechanics. More generally,
$$\mathbf{F}_1 = \sum_n k\frac{q_1q_n}{r_{1n}^2}\hat{\mathbf{r}}_{1n}.$$
If there is no force on a particle, then it is stable. This can be annoying to calculate, so we could look at work/energy instead.