Magnetic susceptibility and relative permeability

[Marwa]

Homework Statement

find the derivation of
χm=μr-1
where Xm is the magnetic susceptibility and μ is the relative permeability.

Homework Equations

M=Xm.H
M is the magnetization,H is the magnetic field
μr=μ/μo
where μo is the permeability of a vacuum,

The Attempt at a Solution

I don't know how to start ,I know some equations related to magnetic properties but i cannot connect them to each other to find the derivation.

 

[collinsmark]

Hello Marwa,

Welcome to Physics Forums!

For convenience, I've retyped in your problem.

Derive the formula,
$$\chi_m = \mu_r - 1$$

Relevant equations:
$$\mathrm{\textbf{M}} = \chi_m \mathrm{\textbf{H}}$$
$$\mu = \mu_r \mu_0$$
I'm pretty sure you will need one or two more relevant equations.
Can you find any relevant equations that relate to the magnetic field, $\mathrm{\textbf{B}}$?
.
.
.

 

[Marwa]

Relevant equations:
β=μH

βo=μoH

μr=μ/μo

β = μoH+ μoM

M=Xm.H

 

[collinsmark]

Relevant equations:
β=μH

βo=μoH

μr=μ/μo

β = μoH+ μoM

M=Xm.H

Okay, with these relevant equations,
$$\mathrm{\textbf{M}} = \chi_m \mathrm{\textbf{H}}$$
$$\mu = \mu_r \mu_0$$
$$\mathrm{\textbf{B}} = \mu \mathrm{\textbf{H}}$$
$$\mathrm{\textbf{B}} = \mu_0 \left( \mathrm{\textbf{H}} +\mathrm{\textbf{M}} \right)$$
you have enough to show that
$$\chi_m = \mu_r - 1.$$
What have you tried so far?

 

[Marwa]

I have tried :

M=Xm.H
Xm=M/H
Xm=Mμ/B
as H=B/μ
and B=μo(H+M)
Xm=(M.μ)/μo(H+M)
so Xm=μr.M/(H+M)

 

[collinsmark]

I have tried :

M=Xm.H
Xm=M/H
Xm=Mμ/B
as H=B/μ
and B=μo(H+M)
Xm=(M.μ)/μo(H+M)
so Xm=μr.M/(H+M)

Try substituting

M = χ_mH​

into

B=μ₀(H+M)​

Then factor the H.

 

[Marwa]

I got it:

B=μo(H+M)
and M=XmH
B=μo(H+XmH)
B=μoH(1+Xm)
so μH=μoH(1+Xm)
μ/μo=μr
Xm=1-μr

Thank you.

 

[Inusa Adamu Gurai]

DERIVE THE RELATIONSHIP B = μ0(H + M)

 

[Arun]

DERIVE THE RELATIONSHIP B = μ0(H + M)

I need the same,
Help me of you find it

 

[collinsmark]

I need the same,
Help me of you find it

Hello @Arun,

I suggest starting a new thread if you need help on this. Be sure to use the automatic template (problem statement, relevant equations, attempted solution, and so forth).