The field lines do, however, pass through the two ends of the cylinder, perpendicular to the ends. No field lines pass through the side of the cylinder. If the plate has a positive charge, the field lines will emerge perpendicular to the plate. To apply Gauss' law, we need to have some idea of what the field looks like. A diagram of the plate, with the Gaussian surface passing through the plate, is shown here: A surface in the form of a box would work, too, but it's got six sides while a cylinder only has three, which simplifies matters a little. In this case, let's work with a cylinder. ![]() With a point charge, the appropriate Gaussian surface was a sphere. To apply Gauss' law, we need to come up with a surface (known as a Gaussian surface) that we can use to apply the law to. In reality, some of the charge will pile up at the edges of the conductor, but we'll assume that the conductor is large enough that these edge effects are negligible. Let's apply Gauss' law to figure out the electric field from a large flat conductor that has a charge Q uniformly distributed over it. So, using Gauss' law we've derived the equation for the field from a point charge. Solving this for the electric field gives: We can put that in place of the sum of the areas to get: If we've reduced that sum to a sum of the area of all the pieces, that's just the total area of the sphere. ![]() We split the sphere up into pieces, work out for each piece, and add them all up. So, what happens with the sum of the areas? Basically what we're doing is this. With a negative charge there'd be a minus sign in front of the E. Because E is constant everywhere over the surface, it can be brought out of the summation sign in Gauss' law. With a positive charge, E and the normal are in opposite directions, giving an angle of 180° and a -1 for the cosine. For a positive charge, the electric field is in the same direction as the normal, the angle between them is 0, and the cosine is 1. In Gauss' law is the angle between the electric field and the line perpendicular to the surface (also known as the normal to the surface), pointing out from the surface. The fact that the field lines are perpendicular to the surface simplifies Gauss' law. The other important thing to realize is that the magnitude of the electric field will be the same at each point on the sphere, because each point is the same distance away from the charge. With a negative charge, the field lines will be directed toward the charge they will still pass through the sphere perpendicular to the surface, however. If the charge is positive, the field lines will radiate out from the charge and pass through the sphere perpendicular to the surface of the sphere. To determine the field a distance r away from the charge, the sphere should have a radius of r, and it should be centered on the charge. To apply Gauss' law, all we need to do is to surround the point charge with an imaginary sphere. This is the answer we should get if we apply Gauss' law. Let's use Gauss' law to calculate the electric field from a point charge of size Q, at a distance r away from the charge. The relationship between the two is this:Ĭalculating the field from a point charge What is the permittivity of free space? It's a constant related to the constant k that appears in Coulomb's law. Gauss' Law - the sum of the electric flux through a surface is equal to the charge enclosed by a surface divided by a constant We'll state the general form, and then apply it to two examples to see how it gets used. Gauss' Law looks a little complicated, but when the geometry of a situation is straightforward, which it will be for any case we consider, the equation reduces to a simpler form. We'll take this idea of electric flux, and, by applying Gauss' Law, we'll use it to figure out the electric field. ![]() The maximum flux occurs when the field is perpendicular to the surface. If the electric field is parallel to the surface, no field lines pass through the surface and the flux will be zero. To calculate the flux through a particular surface, multiply the surface area by the component of the electric field perpendicular to the surface. Electric flux is a measure of the number of electric field lines passing through an area. It can appear complicated, but it's straightforward as long as you have a good understanding of electric flux. Gauss' Law is a powerful method for calculating the electric field from a single charge, or a distribution of charge. They're useful to see, but from this class we'll really expect you to be able to apply basic ideas about electric field rather than use Gauss' Law to derive electric fields. Please note that although these notes deal primarily with Gauss' Law, we're going to downplay the derivations using Gauss' Law.
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