Gauss's Law Concepts
With applications to
Let's explore some of the concepts behind Gauss's Law. A point charge generates an electric field that is the same in every direction. That is, the field is spherically symmetric. What happens if we around the point charge? There is nothing physical there, we simply draw a sphere.
We see all the electric field lines pass through the sphere. The field lines are outward bound for a , and inward bound for a . When the charge is at the of the sphere, the field lines are perpendicular, or normal, to the surface where they cross it. The field lines are also evenly distributed over the surface, which tells us that the electric field has a constant strength at any point on the sphere. If we move the charge to the , to the , if we move it , or if we move it , as long as the charge remains within the surface the same fields lines cross the surface in the same direction. Where the field lines cross the surface changes, and the angle between the field lines and the surface changes, but the same field lines cross the surface no matter where the charge is within the surface.
Take the charge of the surface, and something completely different happens. Look carefully, when the charge is outside of the surface many of the field lines no longer cross the surface at all, and those that do cross the surface pass into and out of the surface. A charge outside of the surface makes no contribution to field lines passing through the surface.
If we the size of the surface we see that the fields lines are spread further apart where they cross the surface. This corresponds to a weaker field. So the electric field gets weaker the further away from the point charge we get. For a surface, we see the opposite effect. The field lines are closer together where they cross the surface, so the field is stronger closer to the point charge.
If instead we , we lose the nice features we we saw when we paired a sphere with a spherically symmetric field. The same field lines cross the surface in the same direction, but they intersect the surface at a wide variety of angles, and their density varies significantly across the surface.
Now we go on to apply Gauss's law to a point charge.