Nichrome wire consists of a nickel-chromium-iron alloy, is commonly used in heating elements such as on a stove, and has conductivity `2.0 xx 10^(6) (Omega.m)^(-1).` If a Nichrome wire with a cross-sectional area of 2.3 mm`""^(2)?` carries a current of 5.5 A when a 1.4 V potential difference is applied between its ends, what is the wire's length?

Answer

To find the length of the Nichrome wire, we can follow these steps:

### Step 1: Understand the relationship between conductivity, resistivity, and resistance
The conductivity (\( \sigma \)) is related to resistivity (\( \rho \)) by the equation:
\[ \sigma = \frac{1}{\rho} \]
Where resistivity (\( \rho \)) can be expressed in terms of resistance (\( R \)), length (\( L \)), and cross-sectional area (\( A \)):
\[ \rho = \frac{R \cdot A}{L} \]

### Step 2: Relate resistance to voltage and current
Using Ohm's law, the resistance can also be expressed as:
\[ R = \frac{V}{I} \]
Where \( V \) is the potential difference and \( I \) is the current.

### Step 3: Substitute resistance into the resistivity equation
Substituting \( R \) into the resistivity equation gives:
\[ \rho = \frac{V \cdot A}{I \cdot L} \]
From this, we can express conductivity in terms of known quantities:
\[ \sigma = \frac{L \cdot I}{V \cdot A} \]

### Step 4: Rearrange to find the length of the wire
Rearranging the equation to solve for \( L \):
\[ L = \frac{\sigma \cdot V \cdot A}{I} \]

### Step 5: Substitute the known values
Given:
- Conductivity \( \sigma = 2.0 \times 10^6 \, \Omega^{-1} \cdot m^{-1} \)
- Voltage \( V = 1.4 \, V \)
- Current \( I = 5.5 \, A \)
- Cross-sectional area \( A = 2.3 \, mm^2 = 2.3 \times 10^{-6} \, m^2 \)

Substituting these values into the equation:
\[ L = \frac{(2.0 \times 10^6) \cdot (1.4) \cdot (2.3 \times 10^{-6})}{5.5} \]

### Step 6: Calculate the length
Calculating the numerator:
\[ (2.0 \times 10^6) \cdot (1.4) \cdot (2.3 \times 10^{-6}) = 6.46 \]
Now, divide by the current:
\[ L = \frac{6.46}{5.5} \approx 1.17 \, m \]

### Final Answer
Thus, the length of the Nichrome wire is approximately:
\[ L \approx 1.17 \, m \]

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