Question:

The relationship between the magnetic susceptibility $ \chi $ and the magnetic permeability $ \mu $ is given by:
$ \mu_0 $ is the permeability of free space and $ \mu_r $ is relative permeability.

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In problems involving magnetic susceptibility, remember that the relative permeability $ \mu_r $ is directly related to $ \chi $, and permeability $ \mu $ is proportional to $ \mu_0 \times \mu_r $.

Updated On: Oct 30, 2025

$ \chi = \frac{\mu}{\mu_0} - 1 $
$ \chi = \frac{\mu + 1}{\mu_0} $
$ \chi = \mu_r + 1 $
$ \chi = 1 - \frac{\mu}{\mu_0} $

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The Correct Option is A

Approach Solution - 1

We have: \[ \mu_r = (1 + \chi) \quad \text{so} \quad \chi = (\mu_r - 1) \] Also, \[ \mu = \mu_0 \mu_r \quad \Rightarrow \quad \mu_r = \frac{\mu}{\mu_0} \] Thus, \[ \chi = \frac{\mu}{\mu_0} - 1 \]

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Approach Solution -2

The problem asks for the correct relationship between magnetic susceptibility ($\chi$) and magnetic permeability ($\mu$), also involving the permeability of free space ($\mu_0$) and relative permeability ($\mu_r$).

Concept Used:

The solution is derived from the fundamental definitions of magnetic quantities.

Magnetic Field Strength ($ \vec{H} $): This is the external magnetic field that magnetizes a material.
Magnetization ($ \vec{M} $): This is the magnetic dipole moment induced per unit volume within the material in response to the external field.
Magnetic Flux Density ($ \vec{B} $): This is the total magnetic field inside the material, which is the sum of the external field and the field due to magnetization. The relationship is: \[ \vec{B} = \mu_0 (\vec{H} + \vec{M}) \]
Magnetic Susceptibility ($ \chi $): It is a dimensionless quantity that describes how a material responds to an applied magnetic field. It is defined as the ratio of magnetization to the magnetic field strength: \[ \vec{M} = \chi \vec{H} \]
Magnetic Permeability ($ \mu $): It is a measure of a material's ability to support the formation of a magnetic field. It is defined as the ratio of magnetic flux density to the magnetic field strength: \[ \vec{B} = \mu \vec{H} \]
Relative Permeability ($ \mu_r $): It is the ratio of the permeability of a medium to the permeability of free space: \[ \mu_r = \frac{\mu}{\mu_0} \]

Step-by-Step Solution:

Step 1: Start with the fundamental relationship between $ \vec{B} $, $ \vec{H} $, and $ \vec{M} $.

\[ \vec{B} = \mu_0 (\vec{H} + \vec{M}) \]

Step 2: Substitute the definition of magnetic susceptibility ($ \vec{M} = \chi \vec{H} $) into the equation from Step 1.

\[ \vec{B} = \mu_0 (\vec{H} + \chi \vec{H}) \]

Step 3: Factor out the magnetic field strength $ \vec{H} $ from the expression.

\[ \vec{B} = \mu_0 (1 + \chi) \vec{H} \]

Step 4: Use the definition of magnetic permeability ($ \vec{B} = \mu \vec{H} $) to relate $ \mu $ and $ \chi $.

By comparing the equation from Step 3 with $ \vec{B} = \mu \vec{H} $, we can equate the expressions for $ \vec{B} $:

\[ \mu \vec{H} = \mu_0 (1 + \chi) \vec{H} \]

Canceling $ \vec{H} $ from both sides, we get the relationship between $ \mu $, $ \mu_0 $, and $ \chi $:

\[ \mu = \mu_0 (1 + \chi) \]

Final Computation & Result:

Step 5: Rearrange the equation to express $ \chi $ in terms of $ \mu $ and $ \mu_0 $.

Divide both sides of the equation by $ \mu_0 $:

\[ \frac{\mu}{\mu_0} = 1 + \chi \]

Now, solve for $ \chi $:

\[ \chi = \frac{\mu}{\mu_0} - 1 \]

We can also express this using the definition of relative permeability, $ \mu_r = \frac{\mu}{\mu_0} $:

\[ \chi = \mu_r - 1 \]

Comparing our derived result with the given options, the correct relationship is:

$ \chi = \frac{\mu}{\mu_0} - 1 $