Which of the following cylindrical rods of the same metal has the highest rate of flow of heat ? The rods have equal difference of temperature between their ends :

Options

A. l = 2m, r=1 cm
B. l=4m,r=2cm
C. l = 2 m, r = 2 cm
D. l = 2 m, r = 4 cm

Explanation

To determine which cylindrical rod of the same metal has the highest rate of flow of heat, we can use the formula for heat transfer through a cylindrical rod:

\[ \frac{dq}{dt} = \frac{k \cdot A \cdot \Delta T}{L} \]

Where:
- \(\frac{dq}{dt}\) is the rate of heat flow,
- \(k\) is the thermal conductivity of the material,
- \(A\) is the cross-sectional area of the rod,
- \(\Delta T\) is the temperature difference between the ends of the rod,
- \(L\) is the length of the rod.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Since all rods are made of the same metal, the thermal conductivity \(k\) is constant for all rods.
- The temperature difference \(\Delta T\) is also the same for all rods as stated in the question.

2. **Express the Rate of Heat Flow**:
- The formula simplifies to:
\[ \frac{dq}{dt} \propto \frac{A}{L} \]
This means that the rate of heat flow is directly proportional to the cross-sectional area \(A\) and inversely proportional to the length \(L\).

3. **Calculate the Cross-Sectional Area for Each Rod**:
- The cross-sectional area \(A\) of a cylindrical rod is given by:
\[ A = \pi r^2 \]
where \(r\) is the radius of the rod.

4. **Evaluate Each Rod**:
- Let’s denote the radius of each rod as follows:
- Rod 1: \(r_1 = 1\) (Area \(A_1 = \pi (1^2) = \pi\))
- Rod 2: \(r_2 = 2\) (Area \(A_2 = \pi (2^2) = 4\pi\))
- Rod 3: \(r_3 = 2\) (Area \(A_3 = \pi (2^2) = 4\pi\))
- Rod 4: \(r_4 = 4\) (Area \(A_4 = \pi (4^2) = 16\pi\))

5. **Consider the Length of Each Rod**:
- Assume the lengths of the rods are as follows:
- Rod 1: \(L_1 = 2\)
- Rod 2: \(L_2 = 4\)
- Rod 3: \(L_3 = 2\)
- Rod 4: \(L_4 = 2\)

6. **Calculate the Rate of Heat Flow for Each Rod**:
- For Rod 1:
\[ \frac{dq}{dt}_1 \propto \frac{\pi}{2} = \frac{1}{2} \]
- For Rod 2:
\[ \frac{dq}{dt}_2 \propto \frac{4\pi}{4} = 1 \]
- For Rod 3:
\[ \frac{dq}{dt}_3 \propto \frac{4\pi}{2} = 2 \]
- For Rod 4:
\[ \frac{dq}{dt}_4 \propto \frac{16\pi}{2} = 8 \]

7. **Compare the Rates of Heat Flow**:
- The calculated rates of heat flow are:
- Rod 1: \(0.5\)
- Rod 2: \(1\)
- Rod 3: \(2\)
- Rod 4: \(8\)

8. **Conclusion**:
- The rod with the highest rate of flow of heat is Rod 4, as it has the highest calculated value of \(\frac{dq}{dt}\).

### Final Answer:
The cylindrical rod with the highest rate of flow of heat is **Rod 4**.