Two rods of equal length and diameter have thermal conductivite 3 and 4 units respectively. If they are joined in series, the thermal conductivity of the combination in the given units would be

Options

A. `3.43`
B. `4.43`
C. `5.43`
D. `2.43`

Explanation

To find the equivalent thermal conductivity of two rods joined in series, we can follow these steps:

### Step 1: Understand the Given Information
We have two rods with the following properties:
- Length of both rods (L1 and L2) = L (equal for both)
- Diameter of both rods = same (implying equal cross-sectional area)
- Thermal conductivity of the first rod (k1) = 3 units
- Thermal conductivity of the second rod (k2) = 4 units

### Step 2: Calculate the Thermal Resistance of Each Rod
The thermal resistance (R) of a rod can be calculated using the formula:
\[ R = \frac{L}{k \cdot A} \]
where:
- \(L\) = length of the rod
- \(k\) = thermal conductivity
- \(A\) = cross-sectional area

For the first rod (R1):
\[ R_1 = \frac{L}{k_1 \cdot A} = \frac{L}{3 \cdot A} \]

For the second rod (R2):
\[ R_2 = \frac{L}{k_2 \cdot A} = \frac{L}{4 \cdot A} \]

### Step 3: Find the Total Thermal Resistance
When two resistances are in series, the total resistance (R_equivalent) is the sum of the individual resistances:
\[ R_{\text{equivalent}} = R_1 + R_2 = \frac{L}{3A} + \frac{L}{4A} \]

To add these fractions, find a common denominator (which is 12A):
\[ R_{\text{equivalent}} = \frac{4L}{12A} + \frac{3L}{12A} = \frac{7L}{12A} \]

### Step 4: Relate Total Resistance to Equivalent Thermal Conductivity
The equivalent thermal resistance can also be expressed in terms of the equivalent thermal conductivity (k_equivalent):
\[ R_{\text{equivalent}} = \frac{L}{k_{\text{equivalent}} \cdot A} \]

Setting the two expressions for \(R_{\text{equivalent}}\) equal to each other:
\[ \frac{L}{k_{\text{equivalent}} \cdot A} = \frac{7L}{12A} \]

### Step 5: Solve for k_equivalent
Cancel \(L\) and \(A\) from both sides:
\[ \frac{1}{k_{\text{equivalent}}} = \frac{7}{12} \]

Taking the reciprocal gives:
\[ k_{\text{equivalent}} = \frac{12}{7} \]

### Step 6: Calculate the Numerical Value
Now, calculate the numerical value:
\[ k_{\text{equivalent}} = \frac{12}{7} \approx 1.714 \text{ units} \]

### Conclusion
The equivalent thermal conductivity of the combination of the two rods joined in series is approximately **1.714 units**.

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