Question

A point charge, \(+Q,\) is located on the \(x\) -axis at \(x=a\), and a second point charge, \(-Q\), is located on the \(x\) -axis at \(x=-a\). A Gaussian surface with radius \(r=2 a\) is centered at the origin. The flux through this Gaussian surface is a) zero. c) less than zero. b) greater than zero. d) none of the above.

Short answer

Answer: a) zero.

Step-by-step solution

The Gaussian surface is a sphere with a radius of \(r=2a\). The positive charge \(+Q\) is located at \(x=a\), which is inside the Gaussian surface. The negative charge \(-Q\) is located at \(x=-a\), which is also inside the Gaussian surface. #Step 2: Calculate the net charge inside the Gaussian surface#

Since both charges are inside the Gaussian surface, the net charge inside the surface is the sum of the positive charge and the negative charge: \((+Q) + (-Q) = 0\). #Step 3: Apply Gauss's Law to determine the electric flux#

Gauss's Law states that the electric flux through a closed surface is equal to the total charge enclosed by the surface, divided by the permittivity of free space, \(\varepsilon_{0}\). The electric flux \(\Phi_E\) is given by: \(\Phi_E = \frac{Q_{enclosed}}{\varepsilon_{0}}\) As we found in Step 2, the net charge inside the Gaussian surface is zero. Therefore, the electric flux through the Gaussian surface is also zero: \(\Phi_E = \frac{0}{\varepsilon_{0}} = 0\) #Step 4: Choose the correct answer#

Based on our calculations, the electric flux through the Gaussian surface is zero. Therefore, the correct answer is: a) zero.