The coefficient of linear expansion of glass is `alpha_g` per `.^(@)C` and the cubical expansion of mercury is `gamma_m` per `.^(@)C`. The volume of the bulb of a mercury thermometer at `0^@C` is `V_0` and cross section of the capillary is `A_0`. What is the length of mercury column in capillary at `T^@C`, if the mercury just fills the bulb at `0^@C` ?

Options

A. `(V_0T(gamma_m+3alpha_g))/(A_0(1+2alpha_gT))`
B. `(V_0T(gamma_m-3alpha_g))/(A_0(1+2alpha_gT))`
C. `(V_0T(gamma_m+2alpha_g))/(A_0(1+3alpha_gT))`
D. `(V_0T(gamma_m-2alpha_g))/(A_0(1+3alpha_gT))`

Explanation

To solve the problem, we need to find the length of the mercury column in the capillary at temperature \( T^\circ C \). We will consider the expansions of both the mercury and the glass bulb.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Coefficient of linear expansion of glass: \( \alpha_g \) (per \( ^\circ C \))
- Cubical expansion of mercury: \( \gamma_m \) (per \( ^\circ C \))
- Volume of the bulb at \( 0^\circ C \): \( V_0 \)
- Cross-section of the capillary: \( A_0 \)

2. **Calculate the Change in Volume of Mercury:**
The change in volume of mercury when the temperature changes from \( 0^\circ C \) to \( T^\circ C \) is given by:
\[ \Delta V_{mercury} = \gamma_m \cdot V_0 \cdot T \]

3. **Calculate the Change in Volume of the Glass Bulb:**
The change in volume of the glass bulb is given by:
\[ \Delta V_{bulb} = 3 \alpha_g \cdot V_0 \cdot T \]
(Note: The factor of 3 comes from the fact that we are dealing with volume expansion, which is three times the linear expansion.)

4. **Determine the Apparent Change in Volume of Mercury:**
The apparent expansion of mercury, which accounts for the expansion of the glass bulb, is:
\[ \Delta V_{apparent} = \Delta V_{mercury} - \Delta V_{bulb} \]
Substituting the previous results:
\[ \Delta V_{apparent} = \gamma_m \cdot V_0 \cdot T - 3 \alpha_g \cdot V_0 \cdot T \]
\[ \Delta V_{apparent} = V_0 \cdot T (\gamma_m - 3 \alpha_g) \]

5. **Calculate the Length of the Mercury Column:**
The length of the mercury column \( H \) in the capillary can be calculated using the formula:
\[ H = \frac{\Delta V_{apparent}}{A_0} \]
Substituting the expression for \( \Delta V_{apparent} \):
\[ H = \frac{V_0 \cdot T (\gamma_m - 3 \alpha_g)}{A_0} \]

6. **Adjust for the Expansion of the Glass Bulb:**
The area of the capillary also expands, which needs to be taken into account. The new area \( A \) at temperature \( T \) is given by:
\[ A = A_0 (1 + 2 \alpha_g T) \]
Therefore, the length of the mercury column becomes:
\[ H = \frac{V_0 \cdot T (\gamma_m - 3 \alpha_g)}{A_0 (1 + 2 \alpha_g T)} \]

### Final Expression:
Thus, the final expression for the length of the mercury column in the capillary at temperature \( T^\circ C \) is:
\[ H = \frac{V_0 \cdot T (\gamma_m - 3 \alpha_g)}{A_0 (1 + 2 \alpha_g T)} \]