To determine the electric flux through the cube and the charge contained within it, we can use Gauss's Law and the concept of electric flux. Let's break this down step by step.
Understanding Electric Flux
Electric flux (Φ) through a surface is defined as the product of the electric field (E) and the area (A) of the surface projected in the direction of the field. Mathematically, it can be expressed as:
Φ = E · A
Where:
- Φ is the electric flux.
- E is the electric field strength.
- A is the area of the surface through which the field lines pass.
Given Parameters
In this scenario, we have a cube with a side length of 16 cm, which is equivalent to 0.16 m. The electric field is directed along the x-axis with a strength of 500 N/C, while the components along the y-axis and z-axis are zero. The left face of the cube is positioned at x = 1600 cm (or 16 m).
Calculating the Area of the Faces
The cube has six faces, but we are particularly interested in the faces that are perpendicular to the electric field. In this case, the faces are the left face (at x = 16 m) and the right face (at x = 16.16 m). The area (A) of each face of the cube can be calculated as follows:
A = side² = (0.16 m)² = 0.0256 m²
Calculating the Electric Flux
Since the electric field is only along the x-axis, we will calculate the flux through the left and right faces of the cube:
Flux through the Left Face
The left face is oriented such that the electric field is entering the cube. Therefore, the flux through the left face (Φ_left) is:
Φ_left = -E · A = -500 N/C · 0.0256 m² = -12.8 N·m²/C
Flux through the Right Face
The right face is oriented such that the electric field is exiting the cube. Thus, the flux through the right face (Φ_right) is:
Φ_right = E · A = 500 N/C · 0.0256 m² = 12.8 N·m²/C
Total Electric Flux through the Cube
The total electric flux (Φ_total) through the cube is the sum of the fluxes through all faces. Since the other four faces are parallel to the electric field and do not contribute to the flux, we can simplify this to:
Φ_total = Φ_right + Φ_left = 12.8 N·m²/C - 12.8 N·m²/C = 0 N·m²/C
Determining the Charge within the Cube
According to Gauss's Law, the total electric flux through a closed surface is equal to the charge (Q) enclosed divided by the permittivity of free space (ε₀):
Φ_total = Q / ε₀
Where ε₀ is approximately 8.85 x 10⁻¹² C²/(N·m²). Since we found that the total electric flux is zero, we can conclude:
0 = Q / (8.85 x 10⁻¹²)
This implies that:
Q = 0 C
Summary
In summary, the electric flux through the cube is zero, and consequently, the charge enclosed within the cube is also zero. This result aligns with the fact that the electric field is uniform and does not create a net flux through the closed surface of the cube.
