To solve the problem, we need to interpret the given graph and use Gauss's law, which relates the electric field $E$ to the enclosed charge $Q$ by the equation:

$E \cdot 4 \pi r^2 = \frac{Q}{\epsilon_0},$

where:

• $E$ is the electric field at a distance $r$,
• $r$ is the radial distance,
• $Q$ is the enclosed charge,
• $\epsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2$).

Observations from the Graph

1. For $r < r_s$ (inside the shell): The electric field is zero, so there is no net charge enclosed inside the shell.
2. For $r > r_s$: The electric field is nonzero, which implies that there is charge on the shell, creating the observed field.

We are tasked to calculate the net charge $Q$ on the shell.

Step-by-Step Solution

1. Electric Field at $r = 2r_s$: From the graph, $E(2r_s) = 12.0 \times 10^7 \, \text{N/C}$.

   Substituting into Gauss's law: $E \cdot 4 \pi r^2 = \frac{Q}{\epsilon_0},$ with $r = 2r_s = 2 \times 4\, \text{cm} = 8 \, \text{cm} = 0.08 \, \text{m}$.

2. Rearrange for $Q$: $Q = E \cdot 4 \pi r^2 \cdot \epsilon_0.$

   Substituting the values: $Q = (12.0 \times 10^7) \cdot 4 \pi \cdot (0.08)^2 \cdot (8.85 \times 10^{-12}).$

3. Calculate $Q$: $Q = (12.0 \times 10^7) \cdot 4 \pi \cdot 0.0064 \cdot (8.85 \times 10^{-12}).$ $Q = (12.0 \times 10^7) \cdot 0.000804 \cdot (8.85 \times 10^{-12}).$ $Q = 8.56 \times 10^{-5} \, \text{C}.$

Thus, the net charge on the shell is approximately $Q \approx 8.56 \, \mu\text{C}$.

Would you like a detailed derivation of any step or a breakdown of the graph analysis?

Related Questions:

1. What happens to the electric field inside a uniformly charged shell?
2. How does the electric field vary outside a spherical charge distribution?
3. What is Gauss's law, and how does it simplify for spherical symmetry?
4. How is the electric field proportional to the distance for spherical distributions?
5. Why is the electric field zero inside a conductor or hollow shell?

Tip: Always ensure consistent units (meters, coulombs, etc.) when solving physics problems to avoid errors!