Taken SP25. Taught by Abigail Crites.
If you had a charge $q$ and an infinite plane, finding the potential everywhere would be very hard. However, if you replace the plane with a $-q$ charge, then it becomes easier. These are image charges. Kirchoff’s Rules There is zero net current in and out of a node: $$\int \mathbf{E} \cdot d\ell = 0.$$
Electric Potential If a sphere has an evenly distributed charge density $\rho$, then we know the sphere is not a conductor because in a conductor, the charge will evenly distribute on the surface because that is the minimum energy configuration. Recall that electric potential over a path from $p_1$ to $p_2$ is $$\phi = -\int_{p_1}^{p_2} \mathbf{E} \cdot ds$$ and it measures the potential energy difference per unit charge between any two points....
Flux can be imagined as the amount of flow going perpendicular to a surface. We can write $$\Phi = \int \mathbf{E} \cdot d\mathbf{a}.$$ Gauss’ Law states that $$\int \mathbf{E} \cdot d \mathbf{a} = \frac{1}{\epsilon_0} \sum_i q_i = \frac{1}{\epsilon_0} \int \rho dV.$$ It turns out that because of the inverse square dependence, a charge outside of a Gaussian surface has zero net electric field flux through that surface. What if we wanted to find the electric field at any point in space outside of a sphere of radius $r$ with surface charge density of $\rho$?...
Electric Potential Energy If you have a charges $q_1$ and $q_2$ infinitely far away, and you want to bring them together so that they are $r_{12}$ apart, work has to be done. Work is $$\int \mathbf{F} \cdot d\mathbf{s} = \int_{\infty}^{r_{12}} -\frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r_{12}^2}dr = \frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r_{12}}.$$ Also, recall that work is path independent. Generalizing for multiple points, we have $$U = \frac{1}{2} \sum_{j=1}^N \sum_{j\neq k}^N \frac{1}{4\pi\epsilon_0} \frac{q_1q_k}{r_{jk}}.$$ The $1/2$ is there because pairs of points are double-counted....
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....