Question

A solenoid of length 0.5 m has a radius of 1 cm and is made up of 'm' number of turns. It carries a current of 5 A. If the magnitude of the magnetic field inside the solenoid is $6.28 \times 10^{-3}$ T, then the value of m is:

Answer 1

Correct Answer: 500

The magnetic field inside a solenoid is given by:

\[ B = \mu_0 n i, \]

where:
- \( B = 6.28 \times 10^{-3} \, \text{T} \) is the magnetic field,
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \) is the permeability of free space,
- \( n = \frac{m}{\ell} \) is the number of turns per unit length,
- \( i = 5 \, \text{A} \) is the current,
- \( \ell = 0.5 \, \text{m} \) is the length of the solenoid.

Step 1: Rearranging the Formula
Substituting the given values:

\[ \mu_0 \left( \frac{m}{\ell} \right) i = B. \]

Rearranging to find \( m \):

\[ m = \frac{B \ell}{\mu_0 i}. \]

Step 2: Substituting the Values
Substituting the given values:

\[ m = \frac{6.28 \times 10^{-3} \times 0.5}{4\pi \times 10^{-7} \times 5}. \]

Simplifying:

\[ m = \frac{6.28 \times 10^{-3} \times 0.5}{12.56 \times 10^{-7}}. \]

Further simplification:

\[ m = \frac{3.14 \times 10^{-3}}{12.56 \times 10^{-7}}. \]

Calculating:

\[ m = 500. \]

Therefore, the value of \( m \) is 500.

Answer 2

Step 1: Given data.
Length of solenoid, l = 0.5 m
Radius of solenoid, r = 1 cm = 0.01 m
Current, I = 5 A
Magnetic field, B = 6.28 × 10⁻³ T
Permeability of free space, μ₀ = 4π × 10⁻⁷ T·m/A

Step 2: Formula for magnetic field inside a solenoid.
The magnetic field inside a solenoid is given by:
\[ B = \mu_0 n I \] where n is the number of turns per unit length:
\[ n = \frac{m}{l} \]
Thus,
\[ B = \mu_0 \frac{m}{l} I \]

Step 3: Substitute known values.
\[ 6.28 \times 10^{-3} = (4\pi \times 10^{-7}) \times \frac{m}{0.5} \times 5 \]
Simplify:
\[ 6.28 \times 10^{-3} = 4\pi \times 10^{-7} \times 10 \times \frac{m}{1} \] \[ 6.28 \times 10^{-3} = 4\pi \times 10^{-6} \, m \]

Step 4: Solve for m.
\[ m = \frac{6.28 \times 10^{-3}}{4\pi \times 10^{-6}} = \frac{6.28}{4\pi} \times 10^3 \] \[ m = \frac{6.28}{12.56} \times 10^3 = 0.5 \times 10^3 = 500 \]

Step 5: Final Answer.
The number of turns on the solenoid is:
\[ \boxed{500} \]

Final Answer: 500