I am having some problem with the formulas for calculating the electric fields of an infinite line of charge and an electric dipole. I don't understand conceptually why they are the way they are. Can someone explain? Any help is appreciated! [Note: K=1/(4*pi*epsilon_o), lambda=linear charge density, p=electric dipole moment, E=electic field, r is the distance] Point charge: E=Kq/r^2 <---the electric field falls off as 1/r^2. Infinite line of charge: E=K(2 lambda)/r <---the field falls off as 1/r, i.e. falls off slower than that of a point charge as you move further away. Electric dipole: E=K(2p)/r^3 (field on axis) E=-Kp/r^3 (field in bisecting plane) <---the field falls off as 1/r^3, i.e. falls off faster than that of a point charge as you move further away. Now, is there any easy way to explain the distance-dependence of the electric field of the electric dipole and infinite line of charge? (the colored part above) Why do the formulas make sense? What actually determines the rate at which the electric field falls off as you move away from the charge? Thank you for explaining!Please Register or Log in to view the hidden image!
There are several ways one can think about the problem. Here are two. 1. The line of electric charge is characterized by a charge density which has units of charge/length. Remember that the electric field is proportional to charge and so it will be proportional to the charge density in this case. Now we can use dimensional analysis to determine the behavior of the electric field with r (we're always looking for charge over length squared, right?). Since the linear charge density already provides one factor of 1/length, we need only one more factor of 1/length to get the dimensions right. Hence E falls of as 1/r. Apply the same story to the dipole. A dipole is characterized by a dipole moment which has units of charge*length, and the electric field will be proportional to the dipole moment. Since we now have an extra factor of length on top, we need three factors of length on the bottom, hence the electric field falls off as 1/r^3. Here's another easy one: what's the field of an infinite plane of charge? Such a plane is characterized by its charge per unit area, and since E is proportional to the charge density we apparently don't need any factors of length. Thus we expect that the electric field doesn't depend on r at all! Of course, this is exactly what one finds by solving the equations in detail. 2. On an intuitive level, it makes sense that an infinite line charge should fall off more slowly than an isolated point charge since it contains an infinite amount of charge. On the other hand, a dipole should fall off more rapidly since it has zero net charge. Think in terms of the superposition principle. In the infinite line charge case you're adding up a lot of similar electric fields, enough (infinite) so that the total field falls off more slowly with distance. In the dipole case you're adding up two electric fields that are nearly equal and opposite, close enough so that the total field falls off more rapidly with distance. Does this help?
Yes, this makes a lot of sense now! Thanks Physics Monkey! A line being infinite means that at a particular point, the electric field will be larger, but not necessary the rate in which in falls will be slower, right? Can you explain a little further on this? Besides, why is the electric field of an infinite plane of charge totally indepedent of the distance to the plane? (while the field of an infinite LINE of charge DOES depend on the distance to it...)
I think you can also use dimensional analysis for this one---the charge density is charge/area. Area has two factors of length. The other way that I think of it is in terms of Gauss's Law. (Presumably you know Gauss's Law by now---the infinite plane and infinite wire are the first calculations that students normally do using Gauss's law.) The electric field due a point of charge can go in all directions---the Gaussian surface that surrounds it is a sphere. The electric field due to an infinite line can't go in all directions, it can only point "out" from the line. Now the Gauussian surface is a cylinder---the same amount of field lines that were going through a sphere now have to go through a cylinder. The electric field depends on 1/r because the surface of the cylinder that the lines see depends on 1/r (i.e. \(A = 2 \pi r \times l\)). The last case is the infinite plane. The Gaussian surface is a matchbox, and the area of that matchbox is independant of the distance away from the infinite plane. The electric field lines are all forced to go in one direction now---they can only point perpendicular to the plane. So, the field strength is dependant on how many different directions the electric field lines can go. In the case of the point charge, the field lines can go in all directions, in the inf line, the field lines can go only out from the wire, and in the plane, the field lines have to go perp to the plane. Hope I haven't done any damage to Physics Monkey's insight!
