Question

A solenoid of r=$1$ cm, L=$50$ cm, N=$300$ carries a current of $1$A. Find magnetic flux through a circular area of radius $0.5$cm, whose centre lies on the axis of solenoid.

Answer 1

Hint: A solenoid is an elementary part that comprises a coil of wire that is covered around a metal core. When a current flows through the solenoid, it assembles a consistent magnetic field. The solenoid core's magnetic field is proportional to the current flowing and the number of turns per unit length of the solenoid. The magnetic flux is the number of lines passing through the solenoid per unit area. It is the product of the magnetic field and the area.
Formula Used:
Magnetic field inside the solenoid is given by
$B = \mu_{o} n I$
Where n is the number of turns per unit length.
 is the current flow.
Magnetic flux ($\phi$) is given by-
$\phi = BA$
A is the area through which a number of lines passed.
B is the magnetic field.

Complete answer:
The following information is given in the question-
Length of solenoid, L=$50$ cm = $0.5$ m
Number of turns, N=$300$
We have to find n: number of turns per unit length.
$n = \dfrac{N}{L} = \dfrac{300}{0.5}=600$
Magnetic field’s formula-
$B = \mu_{o} n I = 4 \pi \times 10^{-7} \times 600 \times 1$
$B=7.54 \times 10^{-4}$ T
r=$0.005$ m
$A = \pi r^{2} = \pi (0.005)^{2} = 7.85 \times 10^{-5} m^{2}$
Magnetic flux
$\phi = BA$
$\phi = 7.54 \times 10^{-4} \times 7.85 \times 10^{-5}$
$\phi=5.918 \times 10^{-8}$ Wb
So, the magnetic flux through a circular area of radius $0.5$ cm is $5.918 \times 10^{-8}$Wb.

Additional Information:
Magnetic field inside solenoid is$B = \mu_{o} n I$ and magnetic field outside solenoid is zero because outside the solenoid, field lines per unit of area is less as we compared with the field lines per unit of area inside solenoid. Magnetic field is almost zero at the outer solenoid.

Note:
In the magnetic field formula, n represents the number of field lines per unit area. One can take it as a number of lines, this will lead to wrong results. The magnetic field inside the solenoid is independent of the distance so, it will be the same everywhere inside the solenoid.