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 6.28×1036.28 \times 10^{-3} T, then the value of m is:

Updated On: Mar 22, 2025
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Correct Answer: 500

Solution and Explanation

The magnetic field inside a solenoid is given by:

B=μ0ni, B = \mu_0 n i,

where:
- B=6.28×103T B = 6.28 \times 10^{-3} \, \text{T} is the magnetic field,
- μ0=4π×107T m/A \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} is the permeability of free space,
- n=m n = \frac{m}{\ell} is the number of turns per unit length,
- i=5A i = 5 \, \text{A} is the current,
- =0.5m \ell = 0.5 \, \text{m} is the length of the solenoid.

Step 1: Rearranging the Formula
Substituting the given values:

μ0(m)i=B. \mu_0 \left( \frac{m}{\ell} \right) i = B.

Rearranging to find m m :

m=Bμ0i. m = \frac{B \ell}{\mu_0 i}.

Step 2: Substituting the Values
Substituting the given values:

m=6.28×103×0.54π×107×5. m = \frac{6.28 \times 10^{-3} \times 0.5}{4\pi \times 10^{-7} \times 5}.

Simplifying:

m=6.28×103×0.512.56×107. m = \frac{6.28 \times 10^{-3} \times 0.5}{12.56 \times 10^{-7}}.

Further simplification:

m=3.14×10312.56×107. m = \frac{3.14 \times 10^{-3}}{12.56 \times 10^{-7}}.

Calculating:

m=500. m = 500.

Therefore, the value of m m is 500.

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