If radius of a solid sohere is decreases to half, keeping density of sphere unchanged, the slope of E-r graph inside the sphere will

Options

A. remain unchanged
B. become two times
C. become four times
D. remain `(1)/(8)` th

Answer

To solve the problem, we need to analyze how the gravitational field inside a solid sphere changes when the radius is decreased to half while keeping the density constant. Let's break this down step by step.

### Step-by-Step Solution:

1. **Understanding the Gravitational Field Inside a Solid Sphere**:
The gravitational field \( E \) inside a solid sphere at a distance \( r \) from the center is given by the formula:
\[ E = \frac{G M(r)}{r^2} \]
where \( M(r) \) is the mass enclosed within radius \( r \).

2. **Finding the Mass Enclosed**:
The mass \( M \) of the sphere can be expressed in terms of density \( \rho \) and volume \( V \):
\[ M = \rho V \]
For a sphere, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Therefore, the mass of the sphere is:
\[ M = \rho \left(\frac{4}{3} \pi R^3\right) \]

3. **Substituting Mass in the Gravitational Field Equation**:
Substituting the expression for mass into the gravitational field equation gives:
\[ E = \frac{G \left(\rho \frac{4}{3} \pi R^3\right)}{r^2} \]

4. **Finding the Slope of the E-r Graph**:
The slope of the \( E-r \) graph can be derived from the gravitational field equation. The slope \( S \) can be expressed as:
\[ S = \frac{G \rho \frac{4}{3} \pi R^3}{r^2} \]
Since we are interested in the slope at a specific radius \( r \), we can simplify this to:
\[ S = \frac{G \rho 4 \pi}{3} \]
Here, we note that the slope depends on \( G \) and \( \rho \) but not on \( R \) or \( r \).

5. **Effect of Decreasing Radius**:
When the radius \( R \) is decreased to half, the density \( \rho \) remains unchanged. Thus, the expression for the slope does not change because it does not depend on \( R \) directly.

6. **Conclusion**:
Since the slope \( S \) is independent of the radius \( R \) and only depends on the constants \( G \) and \( \rho \), the slope of the \( E-r \) graph remains unchanged when the radius is halved.

### Final Answer:
The slope of the \( E-r \) graph inside the sphere will **remain unchanged**.