A wire of length 100 m is tightly wounded on a hollow tube of radius 5mm and length 1 m. A current of 1 A is flowing in the wire. Then magnetic field strength inside the tube will be :-

Options

A. 4 T
B. 4 mT
C. 40 mT
D. 40 T

Answer

To find the magnetic field strength inside a hollow tube with a wire wound around it, we can use the formula for the magnetic field inside a solenoid. Here’s the step-by-step solution:

### Step 1: Identify the parameters
- Length of the wire, \( L_w = 100 \, \text{m} \)
- Radius of the tube, \( r = 5 \, \text{mm} = 5 \times 10^{-3} \, \text{m} \)
- Length of the tube, \( L_t = 1 \, \text{m} \)
- Current flowing through the wire, \( I = 1 \, \text{A} \)

### Step 2: Calculate the number of turns (n)
The number of turns \( n \) can be calculated using the formula:
\[ n = \frac{L_w}{\text{Circumference of the tube}} \]
The circumference of the tube is given by:
\[ \text{Circumference} = 2\pi r = 2\pi (5 \times 10^{-3}) \, \text{m} \]
Now substituting the value:
\[ \text{Circumference} = 2\pi (5 \times 10^{-3}) = 10\pi \times 10^{-3} \, \text{m} \]
Now, substituting this into the formula for \( n \):
\[ n = \frac{100}{10\pi \times 10^{-3}} = \frac{100 \times 10^3}{10\pi} = \frac{10^4}{\pi} \, \text{turns} \]

### Step 3: Calculate the magnetic field (B)
The magnetic field inside a solenoid is given by the formula:
\[ B = \mu_0 n I \]
Where \( \mu_0 \) (the permeability of free space) is approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

Substituting \( n \) into the equation:
\[ B = \mu_0 \left(\frac{10^4}{\pi}\right) I \]
Substituting the values:
\[ B = (4\pi \times 10^{-7}) \left(\frac{10^4}{\pi}\right)(1) \]
The \( \pi \) cancels out:
\[ B = 4 \times 10^{-7} \times 10^4 = 4 \times 10^{-3} \, \text{T} \]

### Step 4: Convert to microtesla
Since \( 1 \, \text{T} = 10^6 \, \mu\text{T} \):
\[ B = 4 \times 10^{-3} \, \text{T} = 4 \times 10^{3} \, \mu\text{T} = 4 \, \mu\text{T} \]

### Final Answer
The magnetic field strength inside the tube is:
\[ \boxed{4 \, \mu\text{T}} \]