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 within a solenoid is expressed as: \[ B = \mu_0 n i, \] where: - \( B = 6.28 \times 10^{-3} \, \text{T} \) represents the magnetic field strength. - \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \) is the magnetic permeability of free space. - \( n = \frac{m}{\ell} \) denotes the number of turns per unit length. - \( i = 5 \, \text{A} \) is the electric current. - \( \ell = 0.5 \, \text{m} \) is the solenoid's length. Step 1: Formula Rearrangement Substitute the given values into the formula: \[ \mu_0 \left( \frac{m}{\ell} \right) i = B. \] Isolate \( m \): \[ m = \frac{B \ell}{\mu_0 i}. \] Step 2: Value Substitution and Calculation Substitute the numerical values: \[ m = \frac{6.28 \times 10^{-3} \times 0.5}{4\pi \times 10^{-7} \times 5}. \] Simplify the expression: \[ m = \frac{6.28 \times 10^{-3} \times 0.5}{12.56 \times 10^{-7}}. \] Further simplification: \[ m = \frac{3.14 \times 10^{-3}}{12.56 \times 10^{-7}}. \] Calculate the final result: \[ m = 500. \] The calculated value for \( m \) is 500.