A long straigth wire of resistance `R`, radius `a` and length `l` carries a constant current `I`.The poynting vector for the wire will be

Options

A. `(IR)/(2pial)`
B. `IR^(2)/(al)`
C. `(I^(2)R)/(al)`
D. `(I^(2)R)/(2pial)`

Explanation

To find the Poynting vector \( \mathbf{S} \) for a long straight wire of resistance \( R \), radius \( a \), and length \( l \) carrying a constant current \( I \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Poynting Vector**:
The Poynting vector \( \mathbf{S} \) represents the power per unit area carried by an electromagnetic wave and is given by the formula:
\[ \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} \]
where \( \mathbf{E} \) is the electric field and \( \mathbf{B} \) is the magnetic field.

2. **Determine the Electric Field \( \mathbf{E} \)**:
For a long straight wire carrying a current \( I \), the electric field \( \mathbf{E} \) at a distance \( r \) from the wire can be calculated using Ohm's law. The potential difference \( V \) across the length \( l \) of the wire is given by:
\[ V = I \cdot R \]
The electric field \( \mathbf{E} \) is then:
\[ \mathbf{E} = \frac{V}{l} = \frac{IR}{l} \]

3. **Determine the Magnetic Field \( \mathbf{B} \)**:
The magnetic field around a long straight wire carrying current \( I \) can be calculated using the Biot-Savart law. At a distance \( r \) from the wire, the magnetic field \( \mathbf{B} \) is given by:
\[ \mathbf{B} = \frac{\mu_0 I}{2 \pi r} \]

4. **Calculate the Poynting Vector \( \mathbf{S} \)**:
Substitute the expressions for \( \mathbf{E} \) and \( \mathbf{B} \) into the Poynting vector formula:
\[ \mathbf{S} = \frac{1}{\mu_0} \left( \frac{IR}{l} \right) \times \left( \frac{\mu_0 I}{2 \pi a} \right) \]
Simplifying this gives:
\[ \mathbf{S} = \frac{IR \cdot I}{2 \pi l} = \frac{I^2 R}{2 \pi l} \]

5. **Final Expression**:
The final expression for the Poynting vector \( \mathbf{S} \) is:
\[ \mathbf{S} = \frac{I^2 R}{2 \pi l} \]