Question

The electric flux through a spherical Gaussian surface of radius \(R\) centered on a charge \(Q\) is \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right) .\) What is the electric flux through a cubic Gaussian surface of side \(R\) centered on the same charge \(Q ?\) a) less than \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\) d) cannot be determined b) more than \(1200 \mathrm{~N} /\left(\mathrm{Cm}^{2}\right)\) from the information given c) equal to \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\)

Short answer

a) less than \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\) b) greater than \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\) c) equal to \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\) d) cannot be determined Answer: c) equal to \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\)

Step-by-step solution

Recall the formula for electric flux

The electric flux (Φ) through a Gaussian surface enclosing a charge Q can be obtained using the formula: \(Φ = \frac{Q}{ε_0}\), where \(ε_0\) is the vacuum permittivity.

Find the charge Q using the given electric flux through spherical surface

We are given the electric flux Φ through the spherical surface as \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\). We can use this to find the charge Q enclosed by the surface using the formula mentioned in Step 1. \(Q = Φ × ε_0\)

Determine the electric flux through the cubic Gaussian surface

Since the charge Q enclosed by both the cubic and spherical Gaussian surfaces is the same, the electric flux Φ through the cubic Gaussian surface will also be given by the same formula: \(Φ_{cubic} = \frac{Q}{ε_0}\)

Compare the electric fluxes

Since the electric flux through both the spherical and cubic Gaussian surfaces is given by the same formula and encloses the same charge Q, the electric fluxes through both surfaces will be equal: \(Φ_{spherical} = Φ_{cubic}\) Hence, the electric flux through a cubic Gaussian surface of side R centered on the same charge Q is equal to \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\). So, the correct answer is: c) equal to \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\)