Step 1: Analyze the electric field inside and outside the sphere.
- Inside the sphere \((r < R)\): Electric field increases linearly with r due to the uniformly distributed charge.
- Outside the sphere \((r ≥ R)\): Electric field decreases as \(\frac{1}{ r^2}\) , behaving like a point charge at the center.
Step 2: Draw the graph.
- Combine the two observations to get the correct graph for the electric field variation.
Final Answer: The correct graph is represented by option (4).
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: