Question

The susceptibility of annealed iron at saturation is 5500. Find the permeability of annealed iron at saturation.

Answer 1

Hint: Magnetic susceptibility is the measure of how much a material will be magnetized when an external magnetic field is applied to it. It is a dimensionless quantity formed by the interaction of electrons and nuclei with the external magnetic field. Mathematically it is the ratio of magnetization M to the applied magnetizing field intensity H. We will substitute the value of susceptibility in the formula of permeability and get to the final answer.

Formula used:
Relation between permeability and susceptibility:
\[\mu = {\mu _0}\left( {1 + x} \right)\]
where $\mu $ is absolute permeability, ${\mu _0}$ is the permeability in free space that is equal to $4\pi \times {10^{ - 7}}\dfrac{N}{{{A^2}}}$ and $x$ is the magnetic susceptibility.

Complete step by step answer:
We are given the susceptibility of annealed iron at saturation, $x = 5500$.Now, we know the relation between permeability and susceptibility as,
\[\mu = {\mu _0}\left( {1 + x} \right)\]
So, we know the value of \[{\mu _0}\] and $x$. Substituting them in the above equation, we get,
\[ \Rightarrow \mu = 4\pi \times {10^{ - 7}} \times \left( {1 + 5500} \right)\dfrac{N}{{{A^2}}}\]
Solving the bracket,
\[ \Rightarrow \mu = 4\pi \times {10^{ - 7}} \times 5501\left( {\dfrac{N}{{{A^2}}}} \right)\]
Simplifying the expression, we get,
\[ \Rightarrow \mu = 12.5714 \times {10^{ - 7}} \times 5501\left( {\dfrac{N}{{{A^2}}}} \right)\]
Carrying out the calculations, we get,
\[ \Rightarrow \mu = 12.5714 \times {10^{ - 7}} \times 5501\left( {\dfrac{N}{{{A^2}}}} \right)\]
\[ \Rightarrow \mu = 69155.4287 \times {10^{ - 7}}\left( {\dfrac{N}{{{A^2}}}} \right)\]
Simplifying the expression further,
\[ \Rightarrow \mu = 6.91554287 \times {10^{ - 3}}\left( {\dfrac{N}{{{A^2}}}} \right)\]
Approximating the value of \[\mu \], we get the permeability of annealed iron as,
 \[\therefore \mu =6.91554287 \times {10^{ - 3}}\left( {\dfrac{N}{{{A^2}}}} \right)\]

Note: One should know the formula for permeability of a material in terms of magnetic susceptibility in order to solve the given problem. We should also know the theory behind the formula and the meaning of the term magnetic susceptibility. Basically magnetic susceptibility tells us whether a material will be attracted or repelled when kept in a magnetic field. We should take care of the calculations while doing such numericals so as to be sure of the final answer. Don’t forget to mention units in the final answer.