91Ó°ÊÓ

Gauss's Law is a fundamental principle that helps us determine the electric field generated by symmetrical charge distributions like infinite planes, lines, or spheres. The law states that the electric field flux through any closed surface is proportional to the total charge enclosed within that surface.

In mathematical terms, Gauss's Law is expressed as: \( \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_\text{enc}}{\varepsilon_0} \). This equation tells us that the integral of the electric field \( \mathbf{E} \) over a closed surface (represented by \( d\mathbf{A} \)) is equal to the enclosed charge \( Q_\text{enc} \) divided by the permittivity of free space \( \varepsilon_0 \).

For an infinite plane with uniform surface charge density \( \sigma \), Gauss's Law simplifies to \( E = \frac{\sigma}{2\varepsilon_0} \). This simplification occurs because the electric field is perpendicular to the surface and has the same magnitude on both sides of the plane, allowing us to consider a Gaussian "pillbox" that simplifies calculation. This result is particularly useful when analyzing fields between, above, or below infinite planar surfaces.

Surface charge density, denoted as \( \sigma \), is the amount of electric charge per unit area on a surface. It is measured in units of coulombs per square meter (C/m²).

In practical scenarios like our exercise, we have two separate planes, each with a different surface charge density. For instance:

• The plane at \( z = 0 \) has a surface charge density of \( 8.00 \text{ nC/m}^2 \).
• The plane at \( z = 2.00 \text{ m} \), a bit offset in space, has a surface charge density of \( 3.00 \text{ nC/m}^2 \).

These values directly influence the electric fields these planes produce due to their uniform charge distribution.

The greater the surface charge density, the stronger the electric field produced. This stems from Gauss's Law, where higher \( \sigma \) values result in higher electric field magnitudes \( E \). Thus, knowing \( \sigma \) helps us predict the strength and direction of the electric field a charged surface will impart onto its surroundings.

The superposition principle allows us to calculate the resultant electric field when multiple sources contribute to it. According to this principle, the net electric field at any point is the vector sum of the electric fields due to the individual charges or charged surfaces.

In the scenario we are discussing, there are two charged planes, one at \( z = 0 \) and another at \( z = 2.00 \, \text{m} \). Each plane creates its electric field, denoted \( E_1 \) and \( E_2 \), respectively:

• For the plane at \( z = 0 \): \( E_1 = 452 \text{ N/C} \).
• For the plane at \( z = 2.00 \text{ m}: \) \( E_2 = 170 \text{ N/C} \).

To find the total electric field at a point located between or outside these planes, we simply add the contributions vectorially.

At \( z = 1.00 \text{ m} \), located between the planes, the electric fields oppose each other. So, \( E_{\text{net}} = E_1 - E_2 = 282 \text{ N/C} \). However, at \( z = 3.00 \text{ m} \), where both fields push in the same direction, \( E_{\text{net}} = E_1 + E_2 = 622 \text{ N/C} \). This principle underscores the ability to handle complex electric field scenarios by breaking them down into simpler parts.