A brass disc and a carbon disc of same radius are assembled alternatively to make a cylindrical conductor . The resistance of the cylinder is independent of the temperature . The ratio of thickness of the brass disc to that of the carbon disc is `[alpha` is temperature coefficient of resistance & Neglect linear expansion ]

Options

A. `|(alpha_(C)rho_(C))/(alpha_(B)rho_(B))|`
B. `|(alpha_(C)rho_(B))/(alpha_(B)rho_(C))|`
C. `|(alpha_(B)rho_(C))/(alpha_(C)rho_(B))|`
D. `|(alpha_(B)rho_(B))/(alpha_(C)rho_(C))|`

Answer

To solve the problem, we need to find the ratio of the thickness of the brass disc (tb) to the thickness of the carbon disc (tc) in a cylindrical conductor made of alternating brass and carbon discs. The resistance of the cylinder is independent of temperature, and we are given that α is the temperature coefficient of resistance.

### Step-by-Step Solution:

1. **Understanding Resistance of Each Material**:
The resistance (R) of a cylindrical conductor can be expressed as:
\[ R = \frac{\rho \cdot L}{A} \]
where \( \rho \) is the resistivity, \( L \) is the length (or thickness in this case), and \( A \) is the cross-sectional area.

2. **Resistance of Brass and Carbon Discs**:
For the brass disc:
\[ R_b = \frac{\rho_b \cdot t_b}{A} \]
For the carbon disc:
\[ R_c = \frac{\rho_c \cdot t_c}{A} \]

3. **Total Resistance in Series**:
The total resistance \( R \) of the cylindrical conductor made of brass and carbon discs in series is:
\[ R = R_b + R_c = \frac{\rho_b \cdot t_b}{A} + \frac{\rho_c \cdot t_c}{A} \]

4. **Temperature Dependence of Resistance**:
The change in resistance with temperature is given by:
\[ \frac{dR}{R} = \alpha \cdot dT \]
Integrating this gives:
\[ \ln \left(\frac{R_2}{R_1}\right) = \alpha \Delta T \]
Thus, the resistance at a new temperature can be expressed as:
\[ R = R_0 (1 + \alpha \Delta T) \]

5. **Setting Up the Condition for Independence of Temperature**:
For the total resistance to be independent of temperature, the coefficient of \( \Delta T \) in the total resistance expression must equal zero:
\[ R = R_{b0}(1 + \alpha_b \Delta T) + R_{c0}(1 + \alpha_c \Delta T) \]
This expands to:
\[ R = (R_{b0} + R_{c0}) + (R_{b0} \alpha_b + R_{c0} \alpha_c) \Delta T \]
For the resistance to be independent of temperature:
\[ R_{b0} \alpha_b + R_{c0} \alpha_c = 0 \]

6. **Substituting the Expressions for Resistance**:
Substitute \( R_{b0} = \frac{\rho_b t_b}{A} \) and \( R_{c0} = \frac{\rho_c t_c}{A} \):
\[ \frac{\rho_b t_b}{A} \alpha_b + \frac{\rho_c t_c}{A} \alpha_c = 0 \]
Simplifying gives:
\[ \rho_b t_b \alpha_b + \rho_c t_c \alpha_c = 0 \]

7. **Finding the Ratio of Thicknesses**:
Rearranging the equation:
\[ \rho_b t_b \alpha_b = -\rho_c t_c \alpha_c \]
Taking the absolute values gives:
\[ \frac{t_b}{t_c} = \frac{\rho_c \alpha_c}{\rho_b \alpha_b} \]

### Final Result:
The ratio of the thickness of the brass disc to the thickness of the carbon disc is:
\[ \frac{t_b}{t_c} = \frac{\rho_c \alpha_c}{\rho_b \alpha_b} \]