A system consists of two concentric spherical shells of radii `a` and `b (b>a)` maintained at temperature `T_1` and `T_2` respectively. The radial rate of flow of heat through substance between the two concentric spherical shells is proportional to

Options

A. b - a
B. `(ab)/(b - a)`
C. `log_e(b/a)`
D. `(b - a)/(ab)`

Answer

To find the radial rate of flow of heat through the substance between two concentric spherical shells of radii \( a \) and \( b \) (where \( b > a \)) maintained at temperatures \( T_1 \) and \( T_2 \) respectively, we can use the concept of heat conduction.

### Step-by-Step Solution:

1. **Understanding Heat Flow**: The rate of heat flow (or heat transfer rate) through a material is given by Fourier's law of heat conduction. For spherical shells, this can be expressed as:
\[ \frac{dQ}{dt} = -k \cdot A \cdot \frac{dT}{dr} \]
where \( k \) is the thermal conductivity, \( A \) is the surface area, and \( \frac{dT}{dr} \) is the temperature gradient.

2. **Surface Area of Spherical Shells**: The surface area \( A \) of a spherical shell at radius \( r \) is given by:
\[ A = 4\pi r^2 \]

3. **Temperature Gradient**: The temperature difference between the two shells is \( T_2 - T_1 \). The average temperature gradient can be approximated as:
\[ \frac{dT}{dr} \approx \frac{T_2 - T_1}{b - a} \]

4. **Applying to the Spherical Shells**: The heat flow through the shells can be integrated over the radial distance from \( a \) to \( b \). However, for a steady-state condition, we can consider the average radius for the area:
\[ \frac{dQ}{dt} = k \cdot 4\pi r^2 \cdot \frac{T_2 - T_1}{b - a} \]
where \( r \) can be taken as the average of \( a \) and \( b \).

5. **Substituting Average Radius**: Taking \( r \) as \( \frac{a + b}{2} \) for simplification, we have:
\[ A = 4\pi \left(\frac{a + b}{2}\right)^2 \]

6. **Final Expression for Heat Flow**: The rate of heat flow becomes:
\[ \frac{dQ}{dt} \propto \frac{(a)(b)(T_2 - T_1)}{b - a} \]
This shows that the radial rate of flow of heat is proportional to the product of the inner and outer radii \( ab \) and the temperature difference \( (T_2 - T_1) \) divided by the difference in radii \( (b - a) \).

### Conclusion:
Thus, the radial rate of flow of heat through the substance between the two concentric spherical shells is proportional to:
\[ \frac{ab (T_2 - T_1)}{b - a} \]