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 step by step, we need to find the required length of the winding wire for the solenoid given the parameters.

### Step 1: Understand the formula for the magnetic field in a solenoid
The magnetic field \( B \) inside a solenoid is given by the formula:
\[ B = \frac{\mu_0 N I}{L} \]
where:
- \( B \) = magnetic field (0.2 T)
- \( \mu_0 \) = permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \))
- \( N \) = number of turns of the solenoid
- \( I \) = current (10 A)
- \( L \) = length of the solenoid (0.8 m)

### Step 2: Rearrange the formula to find \( N \)
We can rearrange the formula to solve for \( N \):
\[ N = \frac{B L}{\mu_0 I} \]

### Step 3: Substitute the known values into the equation
Substituting the known values into the equation:
\[ N = \frac{0.2 \times 0.8}{4\pi \times 10^{-7} \times 10} \]

### Step 4: Calculate \( N \)
Calculating the numerator:
\[ 0.2 \times 0.8 = 0.16 \]
Calculating the denominator:
\[ 4\pi \times 10^{-7} \times 10 = 4\pi \times 10^{-6} \]
Now substituting these values:
\[ N = \frac{0.16}{4\pi \times 10^{-6}} = \frac{0.16}{1.25664 \times 10^{-6}} \approx 127,323.95 \]
So, rounding off, we get:
\[ N \approx 4 \times 10^4 \]

### Step 5: Calculate the total length of the wire
The total length of the wire \( L_w \) used in the solenoid can be calculated using the formula:
\[ L_w = 2\pi r N \]
where \( r \) is the radius of the solenoid (0.03 m).

### Step 6: Substitute values to find \( L_w \)
Substituting the values:
\[ L_w = 2\pi \times 0.03 \times 4 \times 10^4 \]

### Step 7: Calculate \( L_w \)
Calculating:
\[ L_w = 2\pi \times 0.03 \times 4 \times 10^4 \approx 2.4 \times 10^3 \, \text{m} \]

### Final Answer
The required length of the winding wire is:
\[ L_w \approx 2.4 \times 10^3 \, \text{m} \]

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