Two wires of the same dimensions but resistivities `rho_(1)` and `rho_(2)` are connected in series. The equivalent resistivity of the combination is

Options

A. `rho_(1) + rho_(2)`
B. `(rho_(1) +rho_(2))/(2)`
C. `sqrt(rho_(1)rho_(2))`
D. `2(rho_(1)+rho_(2))`

Answer

To find the equivalent resistivity of two wires of the same dimensions but different resistivities connected in series, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Let the resistivities of the two wires be \( \rho_1 \) and \( \rho_2 \).
- Both wires have the same length \( L \) and the same cross-sectional area \( A \).

2. **Calculate the Resistance of Each Wire:**
- The resistance \( R_1 \) of the first wire can be calculated using the formula:
\[ R_1 = \frac{\rho_1 L}{A} \]
- The resistance \( R_2 \) of the second wire can be calculated similarly:
\[ R_2 = \frac{\rho_2 L}{A} \]

3. **Determine the Total Resistance in Series:**
- When resistors are connected in series, the total resistance \( R_{total} \) is the sum of the individual resistances:
\[ R_{total} = R_1 + R_2 \]
- Substituting the expressions for \( R_1 \) and \( R_2 \):
\[ R_{total} = \frac{\rho_1 L}{A} + \frac{\rho_2 L}{A} \]

4. **Factor Out Common Terms:**
- Since both terms share the same length \( L \) and cross-sectional area \( A \), we can factor these out:
\[ R_{total} = \frac{L}{A} (\rho_1 + \rho_2) \]

5. **Calculate the Equivalent Resistivity:**
- The equivalent resistivity \( \rho_{effective} \) can be defined as:
\[ R_{total} = \frac{\rho_{effective} \cdot (2L)}{A} \]
- Setting the two expressions for \( R_{total} \) equal gives:
\[ \frac{\rho_{effective} \cdot (2L)}{A} = \frac{L}{A} (\rho_1 + \rho_2) \]
- Canceling \( \frac{L}{A} \) from both sides:
\[ \rho_{effective} \cdot 2 = \rho_1 + \rho_2 \]

6. **Solve for the Equivalent Resistivity:**
- Dividing both sides by 2:
\[ \rho_{effective} = \frac{\rho_1 + \rho_2}{2} \]

### Final Answer:
The equivalent resistivity of the combination is:
\[ \rho_{effective} = \frac{\rho_1 + \rho_2}{2} \]