Gauss' Law for Infinite Sheets of Charge

[kazukamikaze]

Homework Statement

Three parallel, infinite, insulating planes (sheets) of charge are arranged as shown (see attached image). Note carefully the charge desnitties and distances given. From left to right the charge densities are -3σ, +σ, +σ. How does the magnitude of the electric field at point a compare to the magnitude of the field at point b?

Homework Equations

E = σ/2ε^₀

The Attempt at a Solution

From what I recall from my lecture, the E field of an infinite sheet of charge does not depend on distance. Second, I believe that the field lines look something like what I've drawn in the second attached image. For that reason, I argued the magnitude of A was larger than B as the charge density of the sheet is larger.

 

[Dick]

Homework Statement

Three parallel, infinite, insulating planes (sheets) of charge are arranged as shown (see attached image). Note carefully the charge desnitties and distances given. From left to right the charge densities are -3σ, +σ, +σ. How does the magnitude of the electric field at point a compare to the magnitude of the field at point b?

Homework Equations

E = σ/2ε^₀

The Attempt at a Solution

From what I recall from my lecture, the E field of an infinite sheet of charge does not depend on distance. Second, I believe that the field lines look something like what I've drawn in the second attached image. For that reason, I argued the magnitude of A was larger than B as the charge density of the sheet is larger.

Homework Statement

Homework Equations

The Attempt at a Solution

The field at A is going to be the sum of the fields from all three plates, isn't it?

 

[kazukamikaze]

The field at A is going to be the sum of the fields from all three plates, isn't it?

Based off what one of my classmates told me, the field at the two points should be equal.

The only way I can rationalize these answer is by taking the sum of the three plates. I'm just confused as to why this is done.

 

[Dick]

Based off what one of my classmates told me, the field at the two points should be equal.

The only way I can rationalize these answer is by taking the sum of the three plates. I'm just confused as to why this is done.

Because the plates don't block the fields coming from the other plates, if that's what the confusion is. You could also directly use Gauss' theorem.

 

[HalfLostAndSearching]

Your understanding is correct. According to Gauss' Law for infinite sheets of charge, the electric field is constant and perpendicular to the surface of the sheet. The magnitude of the electric field is given by E = σ/2ε0, where σ is the surface charge density.

In this problem, we have three sheets of charge with different charge densities. At point a, the electric field is due to two sheets, one with a charge density of -3σ and the other with a charge density of +σ. At point b, the electric field is only due to one sheet with a charge density of +σ.

Since the magnitude of the electric field is directly proportional to the charge density, we can conclude that the magnitude of the electric field at point a is larger than the magnitude of the electric field at point b. This is because the total charge density at point a is (-3σ + σ) = -2σ, while at point b it is only +σ.

Therefore, the magnitude of the electric field at point a is twice the magnitude of the electric field at point b. This can also be seen in the electric field lines you have drawn, where the field lines are closer together at point a compared to point b, indicating a stronger electric field.