Electric Fields and Gauss's Law

Electric fields arise when charges exert forces on each other. A positive electric charge repels another positive charge while attracting negative ones. When you move a charged object near uncharged objects made of conductive materials like metals, the electrons want to stay with their own kind so they flow away from where there is charge building up on one side of the metal until it reaches equilibrium. This leaves behind a region of more concentrated charges called the electric field. Just because your hand isn’t touching anything doesn’t necessarily mean you aren’t creating an electrical field around yourself through things like body heat causing water molecules to vibrate faster which creates minute amounts of static electricity. These fields have strength, direction, and magnitude associated with them. They can be pictured using vectors which show both direction and size.

Gauss’s Law is a principle related to these fields with some specific conditions that apply only under certain circumstances such as when all the charge is enclosed by a closed surface that also contains boundary charges. If you draw lines perpendicularly outwards from every point on this surface, they will define what mathematicians call Gaussian surfaces. By connecting points along those lines, you get curves that are boundary curves. Now imagine putting imaginary tiny spheres inside our large sphere; if you connect the center of these little spheres together, you end up with something called Gaussian Surfaces. Suppose we have a volume V(r) containing n total charges q_i, and we surround it with a Gaussian surface S. Gauss’s Law tells us how much electric flux goes across this surface:

[ \phi = E . dS ]

Where (E) is the vector indicating the direction and strength of the electric field, and (dS) represents small pieces of the surface area of the Gaussian surface. Since we know that electric fields are always directed away from positively charged regions and towards negatively charged ones, we see that the electric field vectors are going into any given portion of our Gaussian surface. Therefore, our equation simplifies to:

[ \Phi=-\sum_{i}q_i St(\theta_i)\epsilon]

Here, (St(\theta_i)) comes from the function describing the shape of the Gaussian surface, and depends on its orientation relative to particular directions in space. Finally, (\epsilon) is either (+1) or (-1), depending on whether the charge (q_i) is positive or negative respectively. With this formula, we can calculate the electric flux leaving every tiny piece of our Gaussian surface, summing over all the charges in our volume V(r). Understanding Gauss's Law helps physicists measure how strong electric fields are, since it allows them to determine their strengths based off of known quantities like charge density and distance.