Animated Solution for Physics - Electrostatics: Find out the surface charge density at the intersection of point $X=3\text{ m}$ plane and X-axis, in the region of uniform line charge of $8\text{ nC/m}$ lying along the Z-axis in free space.

Have you ever wondered how different dimensions of charge interact with each other? In the world of electrostatics, we often deal with point charges (zero-dimensional), line charges (one-dimensional), and surface charges (two-dimensional). But what happens when we try to translate the effect of a line charge into an equivalent surface charge? That is exactly the thrilling puzzle we are going to solve today!

This problem is a beautiful exercise in spatial visualization and understanding the deep connections hidden within Gauss's Law. Let's dive in and unravel the physics step by step.

Imagine you are standing in a vast, empty 3D space. Right down the middle, along the Z-axis, runs an infinitely long, incredibly thin wire. This wire is uniformly charged, glowing with a linear charge density of $\lambda =8\text{ nC/m}$.

Now, shift your focus to the X-axis. We are interested in a very specific location: the point where the plane $X=3\text{ m}$ slices through the X-axis. This intersection point, let's call it $P$, sits exactly at the coordinates $(3,0,0)$.

The shortest distance from our glowing line charge on the Z-axis to this point $P$ is simply the distance along the X-axis. Therefore, our radial distance is $r=3\text{ m}$.

Before we can talk about surface charge density, we need to know what the electric field looks like at point $P$. The electric field created by an infinitely long line charge radiates outward symmetrically, like the bristles of a round brush.

The magnitude of this electric field at a radial distance $r$ is given by the classic formula derived from Gauss's Law:

To make our calculations a bit more familiar, we can multiply the numerator and denominator by 2, allowing us to introduce Coulomb's constant, $k=\frac{1}{4\pi {ϵ}_0}$. This transforms our equation into:

This field points directly away from the Z-axis, meaning at point $P$, it points straight along the positive X-axis.

Here is where the problem gets incredibly clever. It asks for the 'surface charge density' at that point. But wait, there is no physical surface there! It's just empty space.

What the question is really asking is this: If we were to replace this electric field with the field produced by a local, infinite conducting surface, what would its surface charge density $\sigma$ need to be?

We know from the boundary conditions of conductors that the electric field just outside a conductor is directly proportional to the local surface charge density:

By equating these two perspectives—the field we actually have from the line charge, and the field we imagine from an equivalent surface charge—we build our logic bridge:

Rearranging this to solve for our target variable, $\sigma$, we get:

Now comes the satisfying part: plugging in the numbers and watching the physics collapse into a single, elegant value. Let's substitute our knowns:

- $k=9\times 10^9\text{ N}\cdot {\text{m}}^2/{\text{C}}^2$ - $\lambda =8\times 10^{-9}\text{ C/m}$ - ${ϵ}_0=8.85\times 10^{-12}{\text{ C}}^2/(\text{N}\cdot {\text{m}}^2)$ - $r=3\text{ m}$

Notice how beautifully the powers of ten interact. The $10^9$ from Coulomb's constant perfectly annihilates the $10^{-9}$ from the nano-coulombs!

To match the format of our multiple-choice options, we need to convert this into nano-coulombs per square meter (${\text{nC/m}}^2$). A nano-coulomb is $10^{-9}$ Coulombs. By shifting our decimal point three places to the left, we adjust our exponent:

And there we have it! Option (a) is our correct answer. But don't just walk away with the number; take a moment to appreciate the physical reality we just uncovered.

If you were to place a massive, grounded metal sheet exactly at the plane $X=3\text{ m}$, the electric field from the line charge would pull electrons within the metal, inducing a surface charge. The magnitude of that induced charge density right at the intersection point would be exactly $0.424{\text{ nC/m}}^2$.

You haven't just solved a math problem; you've predicted how the physical universe would react to a specific geometric setup. Keep visualizing, keep questioning, and the world of physics will always reward you with these beautiful moments of clarity!