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 the hollow tube, we can follow these steps:

### Step 1: Calculate the Number of Turns
The total length of the wire is given as 100 m, and the radius of the tube is 5 mm (which we will convert to meters). The circumference of the tube (which is the length of one turn of the wire) is given by the formula:

\[ \text{Circumference} = 2\pi r \]

Where \( r = 5 \text{ mm} = 5 \times 10^{-3} \text{ m} \).

Calculating the circumference:

\[ \text{Circumference} = 2\pi (5 \times 10^{-3}) = 10\pi \times 10^{-3} \text{ m} \]

Now, we can find the total number of turns \( N \) by dividing the total length of the wire by the circumference:

\[ N = \frac{\text{Total Length of Wire}}{\text{Circumference}} = \frac{100 \text{ m}}{10\pi \times 10^{-3} \text{ m}} = \frac{100}{10\pi \times 10^{-3}} = \frac{100 \times 10^3}{10\pi} = \frac{10^4}{\pi} \]

### Step 2: Calculate the Number of Turns per Unit Length
The length of the tube is given as 1 m. The number of turns per unit length \( n \) is given by:

\[ n = \frac{N}{\text{Length of Tube}} = \frac{N}{1 \text{ m}} = \frac{10^4}{\pi} \]

### Step 3: Calculate the Magnetic Field Strength
The magnetic field strength \( B \) inside a solenoid is given by the formula:

\[ B = \mu_0 n I \]

Where:
- \( \mu_0 = 4\pi \times 10^{-7} \text{ T m/A} \) (permeability of free space)
- \( I = 1 \text{ A} \) (current flowing through the wire)

Substituting the values we have:

\[ B = \mu_0 n I = (4\pi \times 10^{-7}) \left(\frac{10^4}{\pi}\right) (1) \]

The \( \pi \) cancels out:

\[ B = 4 \times 10^{-7} \times 10^4 = 4 \times 10^{-3} \text{ T} = 4 \text{ mT} \]

### Final Answer
The magnetic field strength inside the tube is:

\[ B = 4 \text{ mT} \]