Other Charge Distributions - Fields & Potentials

#Advanced Applications of Charge Distributions, Gauss' Law, and Electric Potential

This section dives into more complex scenarios involving charge distributions, Gauss' Law, and electric potential calculations. It builds upon the foundational concepts covered earlier, so make sure you're comfortable with those before proceeding. Let's get started! 🚀

#Extended Charge Distributions

So far, we've treated charged objects as point charges. Now, we'll tackle situations where charge is spread out over a line, ring, or sheet. The key idea is to break down the total charge Q into tiny pieces dq, each contributing a small electric field dE. Then, we integrate to find the total field E. Here's the general formula:

$$dE = k \frac{dq}{r^2} \hat{r}$$

where r is the radius vector.

#Ring of Charge Example

If x >>> a, the equation simplifies to that of a point charge, which is a great way to check your work.

#Line of Charge Example

First, express dq in terms of dy using the linear charge density λ:

Then, integrate to find the total electric field, using the Pythagorean theorem to find r:

#Gauss' Law for Various Shapes

Gauss' Law is your best friend for finding electric fields due to symmetrical charge distributions. Here's a quick rundown of how to apply it to different...