A winding wire which is used to frame a solenoid can bear a maximum 10 A. current If length of solenoid is 80 cm and its corss sectional raedius is 3 cm , then required length of winding wire is (B =0.2T)

Options

A. `1.2xx10^(2)`m
B. `4.8xx10^(2)m`
C. `2.4 xx10^(3) m`
D. `6xx 10^(3) m`

Answer

To solve the problem, we need to determine the required length of the winding wire for a solenoid given the following parameters:

- Maximum current (I) = 10 A
- Length of the solenoid (L) = 80 cm = 0.8 m
- Cross-sectional radius (r) = 3 cm = 0.03 m
- Magnetic field (B) = 0.2 T

### Step-by-Step Solution:

1. **Understand the Formula for Magnetic Field in a Solenoid**:
The magnetic field (B) inside a solenoid is given by the formula:
\[ B = \mu_0 \cdot n \cdot I \]
where:
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \))
- \( n \) is the number of turns per unit length (turns/m)
- \( I \) is the current in amperes (A)

2. **Rearranging the Formula to Find n**:
Rearranging the formula to solve for \( n \):
\[ n = \frac{B}{\mu_0 \cdot I} \]

3. **Substituting the Known Values**:
Substitute the known values into the equation:
\[ n = \frac{0.2}{4\pi \times 10^{-7} \cdot 10} \]
\[ n = \frac{0.2}{4\pi \times 10^{-6}} = \frac{0.2}{1.25664 \times 10^{-6}} \approx 159.15 \, \text{turns/m} \]

4. **Calculating the Total Number of Turns (N)**:
The total number of turns (N) in the solenoid can be calculated using:
\[ N = n \cdot L \]
where \( L = 0.8 \, \text{m} \):
\[ N = 159.15 \cdot 0.8 \approx 127.32 \, \text{turns} \]

5. **Calculating the Length of the Winding Wire**:
The length of the winding wire (L_wire) can be calculated using the formula for the circumference of the solenoid:
\[ L_{\text{wire}} = N \cdot 2\pi r \]
where \( r = 0.03 \, \text{m} \):
\[ L_{\text{wire}} = 127.32 \cdot 2\pi \cdot 0.03 \]
\[ L_{\text{wire}} = 127.32 \cdot 0.1885 \approx 24.0 \, \text{m} \]

### Final Answer:
The required length of the winding wire is approximately **24.0 meters**.