A point charge of 2.0 µC is at the center of a cubical Gaussian surface 60 cm on edge. What is the net electric flux through the surface?
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To find the net electric flux through the surface, we can use Gauss's law. Gauss's law states that the net electric flux through a closed surface is equal to the charge enclosed by the surface divided by the permittivity of free space (ε₀).
1. Calculate the charge enclosed by the surface:
The problem states that there is a point charge of 2.0 µC at the center of the Gaussian surface. Since the point charge is at the center, it will be enclosed by the surface. Therefore, the charge enclosed by the surface is 2.0 µC.
2. Determine the permittivity of free space:
The permittivity of free space, denoted by ε₀, is a fundamental constant with a value of approximately 8.85 x 10⁻¹² C²/N·m².
3. Calculate the net electric flux through the surface:
The formula to find the net electric flux through the surface is given as:
Net Electric Flux = (Charge Enclosed) / (Permittivity of Free Space)
Plugging in the values, we have:
Net Electric Flux = (2.0 µC) / (8.85 x 10⁻¹² C²/N·m²)
Simplifying the units, we get:
Net Electric Flux = (2.0 x 10⁻⁶ C) / (8.85 x 10⁻¹² C²/N·m²)
Dividing the values, we can find:
Net Electric Flux ≈ 2.26 x 10⁵ N·m²/C
Therefore, the net electric flux through the surface is approximately 2.26 x 10⁵ N·m²/C.
To find the net electric flux through the surface, we can use Gauss's law. Gauss's law states that the electric flux through a closed surface is equal to the charge enclosed by the surface divided by the permittivity of free space (ε₀).
In this case, we have a cubical Gaussian surface with a point charge at its center. The charge enclosed by the surface is equal to the charge of the point charge since the surface completely encloses the charge.
Given:
Charge, q = 2.0 µC (microcoulombs)
Permittivity of free space, ε₀ = 8.85 x 10⁻¹² C²/(N·m²)
Now let's calculate the net electric flux:
Step 1: Calculate the charge enclosed by the surface.
The charge enclosed by the surface is equal to the given charge, which is 2.0 µC.
Step 2: Calculate the net electric flux.
The net electric flux (Φ) is given by the equation: Φ = q / ε₀.
Substituting the values, we get:
Φ = 2.0 µC / (8.85 x 10⁻¹² C²/(N·m²))
Step 3: Simplify the expression.
To simplify the expression, we divide the charge by the permittivity of free space.
Φ = 2.0 µC x (N·m²) / 8.85 x 10⁻¹² C²
Step 4: Convert units if necessary.
Given that 1 µC = 10⁻⁶ C, we can convert the units of charge:
Φ = 2.0 x 10⁻⁶ C x (N·m²) / 8.85 x 10⁻¹² C²
Step 5: Calculate the net electric flux.
Now, we can calculate the net electric flux:
Φ = (2.0 x 10⁻⁶) / (8.85 x 10⁻¹²) N·m²/C²
Using a calculator, we find:
Φ ≈ 2.26 x 10⁶ N·m²/C²
Therefore, the net electric flux through the surface is approximately 2.26 x 10⁶ N·m²/C².