Problem 11 - Fundamentals of Physics

Question

Go Figure \(23-35\) shows a closed Gaussian surface in the shape of a cube of edge length \(2.00 \mathrm{~m},\) with one corner at \(x_{1}=5.00 \mathrm{~m}\), \(y_{1}=4.00 \mathrm{~m} .\) The cube lies in a region where the electric field vector is given by \(\vec{E}=-3.00 \mathrm{i}-4.00 y^{2} \mathrm{j}+3.00 \mathrm{k} \mathrm{N} / \mathrm{C},\) with \(y\) in meters. What is the net charge contained by the cube?

Step-by-Step Solution

Answer

The net charge contained by the cube is zero.

Step 1: Calculate Electric Flux through Cube Faces

The electric flux through a closed surface is given by Gauss's law as the sum of electric flux through each face of the cube. The general formula for electric flux through a surface is \( \Phi = \int \vec{E} \cdot d\vec{A} \). For each face of the cube, we need to integrate the dot product of the electric field \( \vec{E} \) and the area vector \( d\vec{A} \).

Step 2: Evaluate Area Vectors for Cube Faces

For a cube with side length 2 m, the area of each face is \( A = 4 \, \text{m}^2 \). The area vector, \( d\vec{A} \), points perpendicular to the surface: - Front and back faces (xy-plane): \( d\vec{A} = (4 - y^2)\hat{j} \)- Left and right faces (yz-plane): \( -3 \hat{i} \) - Top and bottom faces (xz-plane): \( 3 \hat{k} \)

Step 3: Calculate Flux through Each Face

- **Left and Right Faces:** Across the left face, where \( x = 5 \), and right face where \( x = 7 \), the electric field is uniform and can be solved by multiplying by area.- **Top and Bottom Faces:** With the bottom face at \( y=4 \) and top at \( y=6 \), since \( E_y = -4y^2 \), perform the integral geometrically across length \(-4\) to \(6 \) for \( y \).- **Front and Back Faces:** Calculate similar to left and right faces similar.Perform these calculations to find the flux through each from the same plane set from the cube dimensions.

Step 4: Sum Electric Fluxes to Find Net Flux

Sum the electric flux values from all six faces of the cube. By Gauss's law, the sum of the fluxes will give the total flux through the surface: \(\Phi_{net} = \Phi_{front} + \Phi_{back} + \Phi_{left} + \Phi_{right} + \Phi_{top} + \Phi_{bottom}\)

Step 5: Apply Gauss's Law to Find Net Charge

According to Gauss's Law, the net electric flux through a closed surface is related to the net charge \( Q_{enc} \) enclosed by the surface by \( \Phi_{net} = \frac{Q_{enc}}{\varepsilon_0} \). Rearrange to find the net charge: \( Q_{enc} = \Phi_{net} \cdot \varepsilon_0 \), where \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N} \, \text{m}^2 \).

Key Concepts

Electric Flux

Electric flux is a measure of the electric field passing through a certain area. Imagine it like wind passing through a net. The amount of air (wind) that goes through depends on both the strength of the wind and the size of the net. Similarly, electric flux depends on the strength of the electric field and the angle at which the field lines pass through the surface.
The mathematical expression for electric flux, \( \Phi \), through a surface area \( A \) is given by:

• \( \Phi = \int \vec{E} \cdot d\vec{A} \)

Here, \( \vec{E} \cdot d\vec{A} \) represents the dot product of the electric field \( \vec{E} \) and the area vector \( d\vec{A} \). This dot product means we consider only the component of the electric field that is perpendicular to the surface. When calculating the total electric flux, summing the contributions from all parts of a surface becomes crucial.

Closed Surface

A closed surface is a boundary that encases a volume entirely, like a bubble wrapping around a bunch of soap. In physics, especially electromagnetism, closed surfaces are vital because they allow us to apply Gauss's Law. A cube, as presented in the exercise, serves as a perfect example of a closed surface.
Each face of the cube contributes separately to the total electric flux due to its orientation. For a closed surface, such as this cube, every face needs evaluation. This approach ensures that all the electric field lines entering or exiting the enclosed space are accounted for. The closed surface aids us in determining the net charge inside the volume through Gauss's Law.

Net Charge

The net charge refers to the total charge contained within a closed surface. This is an essential value to determine because it gives insight into the electric state within that surface. Thanks to Gauss's Law, we have a powerful way to connect electric flux and the net charge.
According to Gauss's Law, the net electric flux through a closed surface equals the net charge inside, divided by the permittivity of free space, \( \varepsilon_0 \):

• \( \Phi_{net} = \frac{Q_{enc}}{\varepsilon_0} \)
• Rearranging gives: \( Q_{enc} = \Phi_{net} \cdot \varepsilon_0 \)

The permittivity of free space, \( \varepsilon_0 \), is a constant that helps in calculations involving charges and electric fields in a vacuum. Therefore, by measuring or calculating the net electric flux, we can determine the net charge \( Q_{enc} \) enclosed by the surface.

Gaussian Surface

A Gaussian surface is a theoretical concept used in Gauss's Law to simplify the calculation of electric flux and the net charge enclosed. This entirely imaginary surface helps by enclosing charges symmetrically, allowing easy computation of the electric field's effect.
The choice of Gaussian surface depends on the symmetry of the problem. Common Gaussian shapes include spheres, cylinders, and cubes - like the cube in the exercise. The beauty of using a Gaussian surface lies in its ability to simplify complex electromagnetism problems through easier calculations of flux.

• For example, in a cube, each face becomes a convenient plane to assess the contribution of the electric field.
• The symmetry of the surface and field determines how simply we can apply Gauss's Law.

Overall, by correctly choosing a Gaussian surface around a charge distribution, we directly relate the net electric flux to the enclosed charge.