Three parallel sheets of charge, large enough to be treated...

Three parallel sheets of charge, large enough to be treated as infinite sheets, are perpendicular to the $x$ -axis. Sheet $A$ has surface charge density $\sigma_{A}=+8.00 \mathrm{nC} / \mathrm{m}^{2}$. Sheet $B$ is $4.00 \mathrm{~cm}$ to the right of sheet $A$ and has surface charge density $\sigma_{B}=-4.00 \mathrm{nC} / \mathrm{m}^{2} .$ Sheet $C$ is $4.00 \mathrm{~cm}$ to the right of sheet $B,$ so is $8.00 \mathrm{~cm}$ to the right of sheet $A,$ and has surface charge density $\sigma_{C}=+6.00 \mathrm{nC} / \mathrm{m}^{2}$. What are the magnitude and direction of the resultant electric field at a point that is midway between sheets $B$ and $C,$ or $2.00 \mathrm{~cm}$ from each of these two sheets?

Key Concepts

Infinite Sheet of Charge

An infinite sheet of charge creates a uniform electric field that is independent of the distance from the sheet. The magnitude of the field is given by ?/(2??), where ? is the surface charge density and ?? is the permittivity of free space. This concept is fundamental when analyzing problems involving large planar charged surfaces because the edge effects are negligible, and the field remains constant over space.

Principle of Superposition

The principle of superposition states that when calculating the net electric field from multiple charge distributions, the total field at any given point is the vector sum of the individual fields produced by each charge distribution ignored their interactions. This is crucial for solving problems with multiple charged sheets, as it allows for the independent calculation and then algebraic addition of each field.

Electric Field Direction and Charge Sign

The sign of the surface charge density on a sheet determines the direction of the electric field it produces. A positively charged sheet generates an electric field that points away from the surface, while a negatively charged sheet produces an electric field that points toward the surface. Recognizing and applying this rule is essential when summing vector fields from multiple sources to determine the overall field direction.

Vector Addition of Electric Fields

When multiple electric fields interact, they are added as vectors, taking both magnitude and direction into account. This means that even if individual electric fields are uniform, their resultant direction and magnitude at a point depend on the relative positions and orientations of the charged surfaces. This concept is important in problems where fields from different sources overlap, such as between charged sheets.

Transcript

0:00 Hello everyone.

00:01 In this problem, we're asked to find the electric field at a position that is located between two of three infinite sheets of charge.

00:10 So just to tell you kind of a bit of a background of how you're going to solve this problem, for an infinite sheet, what we usually do is we draw a gauze surface, which is the cylindrical -like imaginary surface, that kind of goes through the sheet of charge.

00:29 And the sheet of charge has a surface density that is what we call sigma.

00:36 And so the sigma tells us how much charge is located over a given area of this surface.

00:42 So when we draw the gaussian surface and we use gaussian law, we know that gausson's law tells us that the integral of the electric field dotted with the area of the gaussian surface is equal to the ankle.

00:56 Close charge divided by the permittivity of free space.

01:00 So that's absolute zero.

01:02 So for our case, what we actually have is that the flux of the electric fields through the sides cancel.

01:09 So there's no net flux over the sides of the cylinder.

01:12 But there is flux over the top and the bottom of the cylinder.

01:17 So for that reason, we get that the area that we're integrating over is 2a or two times the area of the top of the cylinder.

01:25 And this is going to be equal to qn closed over epsilon zero, where qn closed, of course, is sigma.

01:32 So this is going to be equal to sigma times a.

01:37 And so the areas drop out from both sides, and we end up with this relationship where we find that the electric field around that infinite sheet of charge is actually a constant.

01:49 So, you know, we're told all of these distances between the various sheets.

01:54 So now, you know, moving on to our setup, we have three sheets that are separated by four centimeters each.

02:00 So we have sheet a, bit of charge density of eight nanoculums per meter squared.

02:06 That is four centimeters away from sheet b, which has a church density of minus four nanoculums per meter squared, which is four centimeters away from c, which has six nanoculums per meter squared.

02:20 And then we're asked to find the electric field, the magnitude and direction, at a distance two centimeters between, or two centimeters away from both sigma b, or from both sheet b and sheet c, that is halfway between the sheets b and c...

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