Two infinite, uniformly charged, flat nonconducting surfaces are mutually perpendicular. One of the surfaces has a charge distribution of \(+30.0 \mathrm{pC} / \mathrm{m}^{2}\), and the other has a charge distribution of \(-40.0 \mathrm{pC} / \mathrm{m}^{2}\). What is the magnitude of the electric field at any point not on either surface?

Using Gauss' Law, the electric field (E) for an infinite charged plane with a surface charge density (σ) can be given by: E = \(\frac{σ}{2ε_0}\) Here, ε₀ = 8.85 x 10⁻¹² C²/N.m² is the vacuum permittivity. Let's denote the electric field due to the first surface with the positive charge distribution as \(E_1\), and the electric field due to the second surface with negative charge distribution as \(E_2\). Now, we can calculate these fields as: \(E_1 = \frac{+30.0 \times 10^{-12}}{2(8.85 \times 10^{-12})}\), \(E_2 = \frac{-40.0 \times 10^{-12}}{2(8.85 \times 10^{-12})}\)

Since the surfaces are mutually perpendicular, their electric fields will also be perpendicular to each other. We can use the Pythagorean theorem to find the magnitude of the resultant electric field (\(E_{total}\)): \(E_{total} = \sqrt{E_1^2 + E_2^2}\) Calculate the values of \(E_1\) and \(E_2\): \(E_1 \approx \frac{30 \times 10^{-12}}{2(8.85 \times 10^{-12})} \approx 1.69 \,\text{N/C}\), \(E_2 \approx \frac{-40 \times 10^{-12}}{2(8.85 \times 10^{-12})} \approx -2.26 \,\text{N/C}\) Now, substitute these values and compute the magnitude of the resultant electric field: \(E_{total} \approx \sqrt{(1.69 \,\text{N/C})^2 + (-2.26 \,\text{N/C})^2} \approx 2.83 \,\text{N/C}\) Thus, the magnitude of the electric field at any point not on either surface is approximately 2.83 N/C.