The rate of heat flow through a slab is `P_("cond")`. If the slab thickness is doubled, its cross-sectional area is halved, and the temperature difference across it is doubled, then the rate of heat flow becomes

Options

A. `2P_("cond")`
B. `(P_("cond"))/(2)`
C. `P_("cond")`
D. `(P_("cond"))/(8)`

Answer

To find the new rate of heat flow \( P' \) through the slab after the changes in thickness, cross-sectional area, and temperature difference, we can use the formula for heat conduction given by Fourier's law:

\[ P = \frac{K \cdot A \cdot \Delta T}{L} \]

where:
- \( P \) is the rate of heat flow,
- \( K \) is the thermal conductivity,
- \( A \) is the cross-sectional area,
- \( \Delta T \) is the temperature difference,
- \( L \) is the thickness of the slab.

### Step 1: Identify the initial conditions
Let the initial values be:
- Thickness \( L \)
- Cross-sectional area \( A \)
- Temperature difference \( \Delta T \)
- Rate of heat flow \( P_{\text{cond}} = P \)

Thus, we can write:
\[ P = \frac{K \cdot A \cdot \Delta T}{L} \]

### Step 2: Apply the changes to the slab
According to the problem:
- The thickness \( L \) is doubled: \( L' = 2L \)
- The cross-sectional area \( A \) is halved: \( A' = \frac{A}{2} \)
- The temperature difference \( \Delta T \) is doubled: \( \Delta T' = 2\Delta T \)

### Step 3: Substitute the new values into the formula
Now, we can substitute these new values into the heat flow formula to find the new rate of heat flow \( P' \):

\[ P' = \frac{K \cdot A' \cdot \Delta T'}{L'} \]

Substituting the new values:
\[ P' = \frac{K \cdot \left(\frac{A}{2}\right) \cdot (2\Delta T)}{2L} \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ P' = \frac{K \cdot A \cdot \Delta T}{2L} \cdot \frac{2}{2} \]
\[ P' = \frac{K \cdot A \cdot \Delta T}{2L} \]
\[ P' = \frac{1}{2} \cdot \frac{K \cdot A \cdot \Delta T}{L} \]

### Step 5: Relate to the original rate of heat flow
Since \( P = \frac{K \cdot A \cdot \Delta T}{L} \), we can express \( P' \) in terms of \( P \):
\[ P' = \frac{1}{2} P \]

### Conclusion
Thus, the new rate of heat flow after the changes is:
\[ P' = \frac{1}{2} P_{\text{cond}} \]

---