Animated Solution for Physics - Electrostatics: Three infinitely long charge sheets are placed as shown in figure. The electric field at point $P$ is

Imagine you are standing in a space where three infinitely large, flat sheets of charge are placed parallel to each other. These sheets are located at different heights along the $z$-axis: the top sheet is at $z=3a$, the middle sheet is at $z=0$, and the bottom sheet is at $z=-a$.

We are asked to find the net electric field at a specific point $P$, which is located somewhere between the top and middle sheets (i.e., between $z=0$ and $z=3a$).

Before we dive into the calculations, let's recall the fundamental formula for the electric field produced by an infinite sheet of charge. The magnitude of this field is given by:

Notice something fascinating here? The electric field is uniform. It does not depend on how far you are from the sheet! The only thing that matters is the surface charge density $\sigma$.

As for the direction, it's quite intuitive: if the sheet has a positive charge, the field points away from it. If the sheet has a negative charge, the field points towards it.

Now, let's apply this principle to each of the three sheets to find their individual contributions to the electric field at point $P$.

1. Field due to the Top Sheet ($z=3a$): This sheet has a positive charge density of $+\sigma$. Since point $P$ is located below this sheet, the electric field ${\vec{E}}_1$ will point away from it, which means it points downwards (in the $-\hat{k}$ direction).

2. Field due to the Middle Sheet ($z=0$): This sheet has a negative charge density of $-2\sigma$. Point $P$ is located above this sheet. Because the charge is negative, the electric field ${\vec{E}}_2$ will point towards the sheet. This means it also points downwards!

3. Field due to the Bottom Sheet ($z=-a$): This sheet has a negative charge density of $-\sigma$. Point $P$ is located above this sheet as well. Similar to the middle sheet, the electric field ${\vec{E}}_3$ will point towards it, which is again downwards.

By the principle of superposition, the net electric field at point $P$ is simply the vector sum of these three individual fields.

Substituting the values we found:

Since all three fields point in the exact same downward direction, we can just add their magnitudes:

Simplifying the fraction, we get our final answer:

This perfectly matches option (b). It's a beautiful demonstration of how fields from multiple sources can constructively superimpose to create a stronger net field!