Question

A solenoid of $1.5m$ long with an inner diameter of $4cm$ has three layers of winding of $1000$ turns each & carries a current of $2A$. The magnetic flux through a cross-section of the solenoid is nearly

• A. $2.5x{10}^{-7}wb$
• B. $6.31x{10}^{-6}wb$
• C. $2.1x{10}^{-6}wb$
• D. $4.1x{10}^{-5}wb$

Answer 1

The correct option is B

$6.31x{10}^{-6}wb$

Step 1: Given data

Length of a solenoid, $L=1.5mm$

Diameter of the solenoid, $d=4cm=0.04m$

Step 2: Find number of turns

Total number of turns,

$$\begin{gathered} N=3x1000 \\ N=3000 \end{gathered}$$

Number of turns per unit length,

$$\begin{gathered} n=N/L \\ n=3000/1.5 \\ n=2000 \end{gathered}$$

Step 3: Find the cross sectional area

The cross-sectional area of the solenoid

$$\begin{gathered} A=\pi d^2/4 \\ A=(3.14){(0.04)}^2/4 \\ A=1.256x{10}^{-3}{m}^2 \end{gathered}$$

Step 4: Find Magnetic field inside the solenoid

Magnetic field inside the solenoid is

$$\begin{gathered} B={\mu }_{0}nI \\ B=4\pi x{10}^{-7}x2000x2 \\ B=5.024x{10}^{-3}Wb/m^2 \end{gathered}$$

Step 5: Magnetic flux through one loop

Magnetic flux through one loop is

$\varphi =BA$

Thus magnetic flux through $1000$ loops

$$\begin{gathered} \varphi =1000BA \\ \varphi =1000x5.024x{10}^{-3}x1.256x{10}^{-3} \\ \varphi =6.31x{10}^{-6}Wb \end{gathered}$$

Hence, option B is correct.