Question

Four charges are placed in three-dimensional space. The charges have magnitudes \(+3 q,-q,+2 q,\) and \(-7 q .\) If a Gaussian surface encloses all the charges, what will be the electric flux through that surface?

Short answer

Question: Determine the electric flux through a Gaussian surface enclosing charges of +3q, -q, +2q, and -7q. Answer: Electric Flux (Φ_E) = (-1q) / ε₀

Step-by-step solution

Determine the net charge enclosed within the Gaussian surface

To find the net charge enclosed within the Gaussian surface, we add up all the individual charges. So, for this case: Net charge = +3q - q + 2q - 7q

Simplify the net charge equation

Now, let's simplify the equation from Step 1: Net charge = 3q - q + 2q - 7q = (-1)q Therefore, the net charge enclosed within the Gaussian surface is -(1)q.

Apply Gauss's Law

Gauss's Law states that the electric flux through a Gaussian surface (Φ_E) is related to the net charge (Q) inside the surface and the electric constant (ε₀) as follows: Φ_E = Q_enclosed / ε₀ In this exercise, the net charge (Q) is -(1)q.

Calculate the electric flux

We can now substitute our values into Gauss's Law and find the electric flux: Φ_E = (-1q) / ε₀ We can't simplify this expression any further because the question only asked for the electric flux in terms of q. Therefore, Electric Flux (Φ_E) = (-1q) / ε₀