In fair weather, the electric field in the air at a particular location immediatelyabove the Ear

In fair weather, the electric field in the air at a...

Key Concepts

Comparison with Gravitational Force

Comparing the electric force with the gravitational force involves understanding the different dependencies and magnitude scales of these fundamental interactions. While both forces follow an inverse-square law with distance, gravitational force depends on mass and is much weaker on a per-charge basis compared to the electric force, which depends on the magnitude of the charges involved.

Coulomb's Law

Coulomb's law quantifies the electrostatic force between two point charges or charged objects, stating that the force is directly proportional to the product of the charges and inversely proportional to the square of the separation distance. This law is essential for calculating the interaction force when charged bodies, like the Earth and Moon, influence each other.

Gauss's Law

Gauss's law relates the electric flux through a closed surface to the total charge enclosed within the surface. This principle is particularly useful for deriving relationships between the electric field and the surface charge density, especially under conditions of symmetry like those encountered near large charged surfaces.

Electric Field

The electric field is a vector quantity that represents the force per unit charge exerted on a small test charge at a point in space. Understanding its magnitude and direction is crucial as it provides insight into how charges interact in a region, such as near the Earth's surface in fair weather conditions.

Surface Charge Density

Surface charge density is the measure of electric charge per unit area on a surface. It plays a fundamental role in determining the nearby electric field, and through Gauss's law, it directly influences how we calculate the field produced by charged surfaces.

Transcript

00:01 In this question, we're told that in fair weather, the electric field in the air at a particular location above the earth surface is 120 newtons per coulum, and that is directed downwards.

00:14 We're asked to then go ahead and calculate a bunch of things, the first of which is the surface charged density on the ground and comment on whether that would be positive or negative.

00:25 So first of all, let's do a quick little drawing.

00:30 So perhaps this represents the earth.

00:35 And we're told that the electric field is pointing straight down and it's 120 newton's per culum.

00:50 So it looks something like this in terms of the electric field.

00:53 We know that above flat surfaces, the electric field generally is just pointing.

00:58 You know, straight up and down with no x component because of all the cancellations from different parts of the surface.

01:08 So in order to find what the surface charge density is, we need to go ahead and use gauss's law.

01:16 So the gaussian surface that i'm going to choose here is just a cylinder with the top above the surface and the bottom below the surface.

01:26 And i'm not even really going to specify the dimensions and you'll see why in a second.

01:31 So let's go ahead and apply gals's law here.

01:35 So we need the surface integral of e .da and that's going to be equal to the charge enclosed divided by epsilon knot.

01:46 Now in terms of the flux which is the left hand side of the equation, the only side of the cylinder we need to worry about is that top side right.

01:55 The electric field inside a conductor is going to be zero.

01:58 So there's no flux through the bottom of the cylinder.

02:02 And then along the sides of the cylinder, the electric field vectors and the normal vectors to the surface are perpendicular.

02:13 So the only side that has any flux through it is the top side.

02:20 We know that the field vector and the normal vector to the surface are anti -paralleled.

02:29 And so e .da will be equal to negative eda.

02:45 And that's just because they point in opposite directions.

02:49 And then the electric field is going to be the same throughout the entire area of the surface.

02:56 So we can take that out of the surface integral and we're just going to get the integral of d .a.

03:02 Which is of course a.

03:04 And so the left hand side is negative, e times a.

03:12 Now, again, we didn't specify any dimensions, so we can't really calculate what the area is, but that's fine, because what i'm going to do is i'm going to rewrite the charge enclosed as the surface density of charge on the surface times the area, right? that'll give the charge enclosed by the sphere, where the area is the top, the area of the top of the cylinder, right? and so what that does is it allows us to cancel out the area, and it gives us a nice way of calculating surface charge density instead of charge itself, right? so now we've just got to calculate what is e times epsilon not.

03:55 E is 120 newtons per coulom, and the epsilon not is our usual 8 .854 times 10 to the minus 12.

04:08 And so when we multiply that out, we get approximately 1 .06 times 10 to the minus 9 coulomes per meter squared.

04:21 Or you can rewrite that as nanocolomes if you wish.

04:28 Okay, so this is the surface charge density.

04:31 There is a negative out in front.

04:33 And the negative is referring to something physical.

04:36 So the surface charge density here is negative.

04:39 And the way we could know that without doing the calculation is because the electric field is pointing down towards the earth.

04:50 Electric field lines are always going to point, are always going to point towards negative charges, right? so that's how we know that there's a negative surface charge density on the earth here.

05:02 So this is part a.

05:04 For part b, we're asked to imagine that the surface charge density is uniform over the entire plant.

05:10 It and we're asked to find what the charge is on the surface of the earth.

05:16 So we've got the surface charge density and we want to know what the total charge is.

05:25 And so if we want to go from surface charge density to total charge, we need to know what the area is.

05:34 So let's calculate the surface area of the earth...

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