A metallic sphere of diameter D has a cavity of diameter d at its centre. If the sphere is heated, the diameter of the cavity will:

Options

A. Decrease
B. Increase
C. Remain unchanged
D. Decrease if `dlt D//2` and increase if `dgt D//2`

Explanation

To solve the problem, let's analyze the situation step by step:

### Step 1: Understanding the Sphere and Cavity
We have a metallic sphere with an outer diameter \( D \) and a central cavity with a diameter \( d \). The sphere is made of metal, which expands when heated.

**Hint:** Visualize the sphere and the cavity to understand how they relate to each other.

### Step 2: Heating the Sphere
When the metallic sphere is heated, every part of the sphere expands uniformly. This means that the outer diameter \( D \) will increase due to thermal expansion.

**Hint:** Remember that thermal expansion affects all dimensions of solid objects.

### Step 3: Effect on the Cavity
As the outer diameter \( D \) increases, the material of the sphere expands outward. Since the cavity is part of the sphere, the diameter of the cavity \( d \) will also increase. This is because the cavity is surrounded by the expanding material of the sphere.

**Hint:** Consider how the material surrounding the cavity behaves when heated.

### Step 4: Conclusion
Therefore, when the sphere is heated, both the outer diameter \( D \) and the inner diameter \( d \) of the cavity will increase. The expansion occurs uniformly, affecting both the sphere and the cavity.

**Hint:** Think about the concept of expansion at a molecular level; all parts expand, including the cavity.

### Final Answer
The diameter of the cavity will increase when the metallic sphere is heated.