When 230 V is applied across a wire that is 14.1 m long and has a 0.30 mm radius, the magnitude of the current density is `198 xx 10^(8) A//m^(2).` Find the resistivity of the wire.

Answer

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

### Step 1: Identify the given values
- Voltage (V) = 230 V
- Length of the wire (L) = 14.1 m
- Radius of the wire (r) = 0.30 mm = 0.30 x 10^-3 m
- Current density (J) = 198 x 10^8 A/m²

### Step 2: Calculate the cross-sectional area (A) of the wire
The cross-sectional area of a wire can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (0.30 \times 10^{-3})^2 \]
\[ A = \pi (0.09 \times 10^{-6}) \]
\[ A \approx 2.827 \times 10^{-7} \, \text{m}^2 \]

### Step 3: Calculate the current (I) using current density (J)
The relationship between current density (J), current (I), and area (A) is given by:
\[ J = \frac{I}{A} \]
Rearranging this gives:
\[ I = J \times A \]
Substituting the values:
\[ I = (198 \times 10^8) \times (2.827 \times 10^{-7}) \]
\[ I \approx 5.605 \, \text{A} \]

### Step 4: Calculate the resistance (R) using Ohm's law
Using Ohm's law, we know:
\[ V = I \times R \]
Rearranging gives:
\[ R = \frac{V}{I} \]
Substituting the values:
\[ R = \frac{230}{5.605} \]
\[ R \approx 41.0 \, \Omega \]

### Step 5: Calculate the resistivity (ρ) using the formula
The resistivity (ρ) can be calculated using the formula:
\[ \rho = R \frac{A}{L} \]
Substituting the values:
\[ \rho = 41.0 \times \frac{2.827 \times 10^{-7}}{14.1} \]
\[ \rho \approx 0.0823 \times 10^{-8} \, \Omega \cdot m \]

### Final Answer
The resistivity of the wire is approximately:
\[ \rho \approx 0.0823 \times 10^{-8} \, \Omega \cdot m \]

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