Name: Hysteresis: Property of Ferromagnetic Substances
Uploaded: 2019-07-19
Duration: 5 min 6 s
Description: The behaviour of ferromagnets is fascinating. The magnetic field due to a ferromagnet exhibits an interesting phenomenon known as hysteresis. Let’s explore.

Question

A long solenoid with air core has n turns per unit length and carries a current

I

. Using Ampere's circuital law, derive an expression for the magnetic field

\vec{B}

at an interior point on its axis. Write an expression for magnetizing intensity

\vec{H}

in the interior of the solenoid.

Accepted Answer

The above figure shows the sectional view of a long solenoid. At various turns of the solenoid, current comes out of the plane of paper at points marked $⊙$ and enters the plane of paper at points marked $⨂$ To determine the magnetic field $\vec{B}$ at any inside point, consider a rectangular closed path abcd as the Amperean loop. According to Ampere's circuital law,

$\oint \vec{B} . \vec{d l} = μ_{0} \times total current through the loop a b c d$

Now, $\oint \vec{B} . \vec{d l} = \int_{a}^{b} \vec{B} . \vec{d l} + \int_{b}^{c} \vec{B} . \vec{d l} + \int_{c}^{d} \vec{B} . \vec{d l} + \int_{d}^{a} \vec{B} . \vec{d l}$

But

$\int_{b}^{c} \vec{B} . \vec{d l} = \int_{b}^{c} B d l \cos 90 ° = 0$

$\int_{c}^{d} \vec{B} . \vec{d l} = 0$ [ $∴ B = 0$ for outside of solenoid]

$\int_{d}^{a} \vec{B} . \vec{d l} = \int_{b}^{c} B d l \cos 90 ° = 0$

$∴ \oint \vec{B} . \vec{d l} = \int_{a}^{b} \vec{B} . \vec{d l} = \int_{0}^{l} B d l = B l$

Where,

$l =$ length of the side $a b$ of the rectangular loop $a b c d$ .

Let number of turns per unit length of the solenoid $= n$

Then number of turns in length l of the solenoid $= n l$

Thus, the current $I$ of the solenoid threads the loop $a b c d, n l$ times.

$∴$ Total current threading the loop $a b c d, = n l I$

$∴ B l = μ_{0} n l I$

$\Rightarrow B = μ_{0} n I$

The magnetizing intensity $\vec{H}$ in the interior of the solenoid, $\vec{H} = μ_{0} n I$ .

A long solenoid with air core has n turns per unit length and carries a current $I$ . Using Ampere's circuital law, derive an expression for the magnetic field $\vec{B}$ at an interior point on its axis. Write an expression for magnetizing intensity $\vec{H}$ in the interior of the solenoid.

Important Questions on Magnetism

Learn and Prepare for any exam you want !

A long solenoid with air core has n turns per unit length and carries a current I. Using Ampere's circuital law, derive an expression for the magnetic field B→ at an interior point on its axis. Write an expression for magnetizing intensity H→ in the interior of the solenoid.

Important Questions on Magnetism

Learn and Prepare for any exam you want !

A long solenoid with air core has n turns per unit length and carries a current $I$ . Using Ampere's circuital law, derive an expression for the magnetic field $\vec{B}$ at an interior point on its axis. Write an expression for magnetizing intensity $\vec{H}$ in the interior of the solenoid.