A point charge of 1.8uC (microcoulumbs) is at the center of a Gaussian cube 55 cm on edge. What is the net electric flux through the surface?
Calculus? Hmmmmm.
Isn't the flux coming out equal to the charge in (divided by permittivity)?
Exactly what is Gauss' Law? http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html
To find the net electric flux through the surface of the Gaussian cube, we can use Gauss's Law. Gauss's Law states that the electric flux through a closed surface is directly proportional to the total charge enclosed by that surface.
The formula to calculate the electric flux, Φ, is given by:
Φ = Q / ε₀
Where:
- Φ is the electric flux
- Q is the total charge enclosed by the Gaussian surface
- ε₀ is the electrical permittivity of free space
In this case, we have a point charge of 1.8 μC at the center of the cube.
To find the net electric flux through the surface, we need to calculate the electric field at any point on the surface. The electric field, E, is defined as the force experienced by a unit positive charge at that point. The electric field of a point charge is given by:
E = k * (Q / r²)
Where:
- E is the electric field strength
- k is Coulomb's constant (8.99 x 10^9 N m²/C²)
- Q is the charge of the point charge (in this case, 1.8 μC)
- r is the distance between the point charge and the point at which we want to find the electric field
Now, let's proceed with the calculation.
Step 1: Calculate the electric field at any point on the surface of the Gaussian cube.
Since the charge is at the center of the cube, the distance between the charge and any point on the surface is equal to half of the diagonal of the cube.
The diagonal of a cube can be calculated using the Pythagorean theorem:
diagonal = √(edge² + edge² + edge²)
= √(55 cm)² + (55 cm)² + (55 cm)²)
= √(3 * 55 cm)²
= 3 * 55 cm
= 165 cm
Since the charge is at the center of the cube, the distance from the charge to the surface is half of the diagonal:
r = 0.5 * 165 cm
= 82.5 cm
Now we can calculate the electric field at any point on the surface using the formula:
E = k * (Q / r²)
= (8.99 x 10^9 N m²/C²) * (1.8 x 10^-6 C) / (0.825 m)²
= (8.99 x 10^9 N m²/C²) * (1.8 x 10^-6 C) / (0.825 m)²
≈ 3.17 x 10^4 N/C (rounded to two significant figures)
Step 2: Calculate the total charge enclosed by the Gaussian surface.
Since there is only one point charge inside the cube, the total charge enclosed by the surface is equal to the charge of the point charge:
Q = 1.8 μC
Step 3: Calculate the electric flux through the surface.
Using Gauss's Law, we can calculate the electric flux:
Φ = Q / ε₀
= (1.8 x 10^-6 C) / (8.85 x 10^-12 C²/N m²)
≈ 2.03 x 10^5 N m²/C (rounded to two significant figures)
Hence, the net electric flux through the surface of the Gaussian cube is approximately 2.03 x 10^5 N m²/C.