An infinite sheet of charge has a uniform charge density of 18.0 microcoulombs per square meter. What is the magnitude of the electric field due to this charge at a point just above the surface of the sheet?
If we sketch out that sheet, we know our sketch doesn’t quite capture the whole picture. Because the sheet is infinitely large, it goes infinitely far out in every direction along this plane. So just imagine this sheet going as far as your eye can see and even farther.
We’re told moreover that this is a charged sheet; it has a uniform charge density on it. We’ll refer to as 𝜎. 𝜎 tells us just how much charge there is per square meter of sheet. If we look at our sheet from the side edge on, then the question asks, “What is the magnitude of the electric field at a point that’s just above the surface of the sheet?”
We can label the electric field at this point 𝐸. And to solve for it, we can realize that there are positive charges uniformly spread throughout this infinite sheet, all of which create electric field lines which contribute to the field at the point we’re interested in.
Because there are infinitely many charges to consider in this infinite charged sheet, it would be nearly impossible to use Coulomb’s law to calculate this electric field on a point-by-point basis. But we can do better.
We can recall a mathematical relationship, which sums up the effects of all of these infinite charges in this sheet. This equation tells us that the electric field created by an infinite sheet of charge is equal simply to the charge density on the sheet divided by two times 𝜖 nought, the permittivity of free space.
Notice what’s not in this equation, there is no indication of distance at all. So this means that when we selected our point just above the surface from an electric field perspective, we might as well selected a point anywhere above the surface. Since the sheet itself is infinite, the distance we are from it doesn’t make a difference.
So let’s move ahead with calculating this electric field 𝐸. We see we’ve been given 𝜎, the surface charge density on the sheet. So all we need to know is 𝜖 nought, which is a constant. We can treat this value as exactly 8.85 times 10 to the negative 12th farads per meter.
When we enter these values into our equation, we’re careful to write 𝜎, our surface charge density, as 18.0 times 10 to the negative sixth coulombs per square meter. With this more helpful notation, we’re ready to calculate 𝐸, the electric field magnitude.
To three significant figures, it’s 1.02 times 10 to the sixth newtons per coulomb. That’s the electric field magnitude created by this infinite sheet. And note that this electric field magnitude holds true for any distance from the sheet.