The specific resistance of manganin is `50 xx 10^(-8) ohm xx m`. The resistance of a cube of length 50 cm will be

Options

A. `10^(-6)` ohm
B. `2.5 xx 10^(-5)` ohm
C. `10^(-8)` ohm
D. `5 xx 10^(-4)` ohm

Answer

To find the resistance of a cube of manganin with a specific resistance of \(50 \times 10^{-8} \, \Omega \cdot m\) and a length of 50 cm, we can follow these steps:

### Step 1: Convert the length from cm to meters
The length of the cube is given as 50 cm. To convert this to meters:
\[ \text{Length} = 50 \, \text{cm} = 0.5 \, \text{m} \]

**Hint:** Remember that 1 cm = 0.01 m, so to convert cm to m, you multiply by 0.01.

### Step 2: Calculate the cross-sectional area of the cube
Since the cube has equal sides, the cross-sectional area \(A\) can be calculated using the formula for the area of a square:
\[ A = \text{side}^2 \]
For a cube with a side length of 0.5 m:
\[ A = (0.5 \, \text{m})^2 = 0.25 \, \text{m}^2 \]

**Hint:** The area of a square is calculated as the side length squared.

### Step 3: Use the formula for resistance
The resistance \(R\) of a material can be calculated using the formula:
\[ R = \frac{\rho \cdot L}{A} \]
where:
- \(R\) is the resistance,
- \(\rho\) is the specific resistance (given as \(50 \times 10^{-8} \, \Omega \cdot m\)),
- \(L\) is the length (0.5 m),
- \(A\) is the cross-sectional area (0.25 m²).

Substituting the values into the formula:
\[ R = \frac{50 \times 10^{-8} \, \Omega \cdot m \cdot 0.5 \, m}{0.25 \, m^2} \]

### Step 4: Simplify the expression
Calculating the numerator:
\[ 50 \times 10^{-8} \cdot 0.5 = 25 \times 10^{-8} \, \Omega \cdot m \]
Now substituting into the resistance formula:
\[ R = \frac{25 \times 10^{-8}}{0.25} \]
Calculating the division:
\[ R = 25 \times 10^{-8} \cdot 4 = 100 \times 10^{-8} \, \Omega = 1 \times 10^{-6} \, \Omega \]

### Final Answer
The resistance of the cube is:
\[ R = 1 \times 10^{-6} \, \Omega \]

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