Which one of the following graphs represents correctly represent the variation of the gravitational field (E) with the distance (r) from the centre of a spherical shell of mass M radius R ?

Options

A.
B.
C.
D.

Explanation

To solve the question regarding the variation of the gravitational field (E) with the distance (r) from the center of a spherical shell of mass M and radius R, we can follow these steps:

### Step 1: Understand the Gravitational Field Inside the Shell
According to the shell theorem, the gravitational field inside a uniform spherical shell of mass is zero. This means that for any point inside the shell (where r < R), the gravitational field E = 0.

### Step 2: Analyze the Gravitational Field Outside the Shell
For points outside the spherical shell (where r > R), the gravitational field behaves as if all the mass of the shell were concentrated at its center. The gravitational field E outside the shell can be expressed using Newton's law of gravitation:
\[ E = \frac{GM}{r^2} \]
where G is the gravitational constant, M is the mass of the shell, and r is the distance from the center of the shell.

### Step 3: Sketch the Graph
- For r < R (inside the shell), E = 0. This means the graph will be flat at zero for this range.
- For r = R, the graph will still be at zero.
- For r > R, E will decrease with the square of the distance (inversely proportional to \( r^2 \)). This means the graph will start to decline as r increases.

### Step 4: Identify the Correct Graph
Based on the above analysis, the correct graph will show:
- A horizontal line at E = 0 for r < R.
- A curve that starts from E = 0 at r = R and decreases as r increases for r > R.

### Conclusion
The correct option that represents this variation is option D.

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