Which of the following circular rods (given radius `r` and length `l`) each made of the same material and whose ends are maintained at the same temperature will conduct most heat?

Options

A. `r = 2r_(0), l = 2l_(0)`
B. `r = 2r_(0), l = l_(0)`
C. `r = r_(0), l = l_(0)`
D. `r = r_(0), l = 2l_(0)`

Explanation

To determine which of the circular rods will conduct the most heat, we can use the formula for heat conduction through a rod. The rate of heat transfer (dQ/dt) through a rod is given by:

\[ \frac{dQ}{dt} = \frac{kA \Delta T}{L} \]

Where:
- \(dQ/dt\) is the rate of heat transfer,
- \(k\) is the thermal conductivity of the material,
- \(A\) is the cross-sectional area of the rod,
- \(\Delta T\) is the temperature difference across the ends of the rod,
- \(L\) is the length of the rod.

Since all rods are made of the same material and have their ends maintained at the same temperature, we can simplify our analysis by focusing on the parameters \(A\) (cross-sectional area) and \(L\) (length).

### Step 1: Determine the Cross-Sectional Area
For a circular rod with radius \(r\), the cross-sectional area \(A\) is given by:

\[ A = \pi r^2 \]

### Step 2: Analyze the Length
The length \(L\) of the rods is given as \(l\).

### Step 3: Substitute into the Heat Transfer Equation
Substituting the expression for \(A\) into the heat transfer equation, we have:

\[ \frac{dQ}{dt} = \frac{k(\pi r^2) \Delta T}{l} \]

### Step 4: Identify Factors Affecting Heat Transfer
From the equation, we can see that the rate of heat transfer is directly proportional to the cross-sectional area \(A\) (which depends on \(r\)) and inversely proportional to the length \(L\).

### Step 5: Compare Different Rods
To determine which rod conducts the most heat, we need to compare the values of \(A\) and \(L\) for each rod. If the radius \(r\) of the rods varies, then the rod with the largest radius will have the largest cross-sectional area and thus will conduct the most heat, assuming all rods have the same length \(l\).

### Conclusion
Therefore, the rod with the largest radius \(r\) will conduct the most heat.