Question

A point charge causes an electric flux of \( -2 \times 10^4 \, \text{Nm}^2\text{C}^{-1} \) to pass through a spherical Gaussian surface of 8.0 cm radius, centered on the charge. The value of the point charge is:

• A. \( 17.7 \times 10^{-7} \, \text{C} \)
• B. \( 15.7 \times 10^{-7} \, \text{C} \)
• C. \( 17.7 \times 10^{-6} \, \text{C} \)
• D. \( 15.7 \times 10^{-6} \, \text{C} \)

Hint:

Gauss's law relates the electric flux through a surface to the charge enclosed by that surface. Be sure to use the correct value for the permittivity of free space (\( \epsilon_0 \)) in calculations.

Answer 1

The Correct Option is A

According to Gauss's law, the electric flux through a closed surface is related to the charge enclosed by the surface: \[ \Phi_E = \frac{q}{\epsilon_0} \] where \( \Phi_E \) is the electric flux, \( q \) is the charge, and \( \epsilon_0 \) is the permittivity of free space. Given that \( \Phi_E = -2 \times 10^4 \, \text{Nm}^2\text{C}^{-1} \) and the radius of the Gaussian surface is \( r = 8.0 \, \text{cm} \), we can solve for the charge \( q \) as: \[ q = \Phi_E \times \epsilon_0 = (-2 \times 10^4) \times (8.85 \times 10^{-12}) = 17.7 \times 10^{-7} \, \text{C} \]

Answer 2

Step 1: Understanding the problem
We are given the electric flux \( \Phi_E = -2 \times 10^4 \, \text{Nm}^2\text{C}^{-1} \) through a spherical Gaussian surface of radius \( r = 8.0 \, \text{cm} = 0.08 \, \text{m} \).
We are asked to find the value of the point charge enclosed by the surface.

Step 2: Use Gauss’s law
According to Gauss’s law, the total electric flux through a closed surface is given by:
\[ \Phi_E = \frac{q}{\varepsilon_0} \] where \( q \) is the total charge enclosed by the surface and \( \varepsilon_0 = 8.854 \times 10^{-12} \, \text{C}^2\text{N}^{-1}\text{m}^{-2} \) is the permittivity of free space.

Step 3: Rearrange for charge
\[ q = \Phi_E \, \varepsilon_0 \] Substitute the given values:
\[ q = (-2 \times 10^4) \times (8.854 \times 10^{-12}) \] \[ q = -1.7708 \times 10^{-7} \, \text{C} \] The negative sign indicates that the charge enclosed is negative. The magnitude of the charge is:
\[ |q| = 1.77 \times 10^{-7} \, \text{C} \]

Step 4: Interpretation of the result
Since the flux is negative, it means that the electric field lines are entering the Gaussian surface, implying that the enclosed charge is negative. The problem, however, asks for the value of the charge, so we take the magnitude.

Final Answer:
\[ \boxed{17.7 \times 10^{-7} \, \text{C}} \]