`A` and `B` are two points on uniform metal ring whose centre is `O` The angle `AOB =theta` A and `B` are maintaind at two different constant temperatures When `theta =180^(@)` the rate of total heat flow from `A` to `B` is `1.2W` When `theta =90^(@)` this rate will be .

Options

A. `0.6 W`
B. `0.9 W`
C. `1.6 W`
D. `1.8 W`

Answer

To solve the problem, we need to analyze the heat flow between two points A and B on a uniform metal ring, considering the change in the angle θ between them.

### Step-by-Step Solution:

1. **Understanding Heat Flow**: The rate of heat flow (Q/t) between two points is given by the formula:
\[ \frac{dQ}{dt} = \frac{(T_A - T_B)}{R_{eq}} \]
where \(T_A\) and \(T_B\) are the temperatures at points A and B, and \(R_{eq}\) is the equivalent resistance between the two points.

2. **Resistance and Angle Relation**: The resistance of the arc between points A and B on the ring depends on the angle θ subtended at the center O. The resistance \(R\) is proportional to the length of the arc, which can be expressed as:
\[ R \propto R \cdot \theta \]
where R is the radius of the ring.

3. **Case 1: θ = 180°**: When θ = 180°, the resistance can be expressed as:
\[ R_{eq1} = k \cdot \frac{180^\circ}{360^\circ} = \frac{k}{2} \]
where k is a constant that includes material properties and dimensions.

4. **Given Heat Flow Rate**: We know that when θ = 180°, the rate of heat flow is given as:
\[ \frac{dQ}{dt} = 1.2 \, W \]

5. **Case 2: θ = 90°**: For θ = 90°, the equivalent resistance becomes:
\[ R_{eq2} = k \cdot \frac{90^\circ}{360^\circ} = \frac{k}{4} \]

6. **Comparing Heat Flow Rates**: Since the temperature difference remains constant, we can set up the ratio of heat flow rates for the two cases:
\[ \frac{\frac{dQ}{dt} \bigg|_{90^\circ}}{\frac{dQ}{dt} \bigg|_{180^\circ}} = \frac{R_{eq1}}{R_{eq2}} \]
Substituting the known values:
\[ \frac{\frac{dQ}{dt} \bigg|_{90^\circ}}{1.2} = \frac{\frac{k}{2}}{\frac{k}{4}} = \frac{4}{2} = 2 \]

7. **Solving for Heat Flow Rate at 90°**:
\[ \frac{dQ}{dt} \bigg|_{90^\circ} = 2 \cdot 1.2 = 2.4 \, W \]

### Final Answer:
The rate of heat flow from A to B when θ = 90° is **2.4 W**.