The radius of a ring is R and its coefficient of linear expansion is `alpha.` If the temperature of ring increases by `theta` then its circumfrence will increase by

Options

A. `pi R alpha theta`
B. `2 pi R alpha theta`
C. `piRalpha(theta)/2`
D. `pi R alpha(theta)/4`

Explanation

To find the increase in the circumference of a ring when its temperature increases, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Linear Expansion**:
The linear expansion of a material is given by the formula:
\[ \Delta L = \alpha \cdot \Delta T \cdot L \]
where:
- \(\Delta L\) is the change in length,
- \(\alpha\) is the coefficient of linear expansion,
- \(\Delta T\) is the change in temperature,
- \(L\) is the original length.

2. **Identify the Original Length**:
For a ring with radius \(R\), the original length (circumference) is:
\[ L = 2\pi R \]

3. **Substitute the Original Length into the Expansion Formula**:
Now, substituting \(L\) into the linear expansion formula:
\[ \Delta L = \alpha \cdot \Delta T \cdot (2\pi R) \]

4. **Express the Change in Circumference**:
Here, \(\Delta T\) is the increase in temperature, denoted as \(\theta\). Thus, we can rewrite the equation as:
\[ \Delta L = 2\pi R \cdot \alpha \cdot \theta \]

5. **Conclusion**:
The increase in the circumference of the ring due to the temperature increase is:
\[ \Delta L = 2\pi R \alpha \theta \]

### Final Answer:
The increase in circumference is \(2\pi R \alpha \theta\).