A copper disc and a carbon disc of same radius are assembled alternately and co-axially to make a cylindrical conductor whose temperature coefficient conductor whose temperature coefficient of resistance is almost equal to zero. Ratio of thickness of the copper and the carbon disc is (neglect change in length. `alpha_(CU)` and `-alpha_(C)` represent the temperature coefficients of resistivities and `p_(cu)` carbon at room temperature respectively)

Options

A. `p_(C)alpha_(C)//p_(CU)alpha_(CU)`
B. `p_(CU)alpha_(CU)//p_(C)alpha_(C)`
C. `p_(C)alpha_(CU)//p_(CU)alpha_(C)`
D. `p_(CU)alpha_(C)//p_(C)alpha_(CU)`

Answer

To solve the problem, we need to determine the ratio of the thickness of the copper disc to the carbon disc in a cylindrical conductor made of these materials. The key point is that the temperature coefficient of resistance of the combined cylindrical conductor is almost equal to zero.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a copper disc and a carbon disc of the same radius, assembled alternately to form a cylindrical conductor. We need to find the ratio of their thicknesses, given that the overall temperature coefficient of resistance is approximately zero.

2. **Temperature Coefficient of Resistance**:
The temperature coefficient of resistance for copper is denoted as \( \alpha_{Cu} \) and for carbon as \( -\alpha_{C} \). The negative sign for carbon indicates that its resistance decreases with increasing temperature.

3. **Condition for Zero Temperature Coefficient**:
For the overall temperature coefficient of resistance to be zero, the contributions from both materials must balance each other out. Mathematically, this can be expressed as:
\[ \alpha_{Cu} \cdot t_{Cu} + (-\alpha_{C}) \cdot t_{C} = 0 \]
where \( t_{Cu} \) and \( t_{C} \) are the thicknesses of the copper and carbon discs, respectively.

4. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ \alpha_{Cu} \cdot t_{Cu} = \alpha_{C} \cdot t_{C} \]

5. **Finding the Ratio**:
We can express the ratio of the thicknesses as:
\[ \frac{t_{Cu}}{t_{C}} = \frac{\alpha_{C}}{\alpha_{Cu}} \]

6. **Final Expression**:
Thus, the ratio of the thickness of the copper disc to the carbon disc is:
\[ \frac{t_{Cu}}{t_{C}} = \frac{\alpha_{C}}{\alpha_{Cu}} \]