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.
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.
Show Hint
- \( \chi = \frac{\mu}{\mu_0} - 1 \)
- \( \chi = \frac{\mu + 1}{\mu_0} \)
- \( \chi = \mu_r + 1 \)
- \( \chi = 1 - \frac{\mu}{\mu_0} \)
The Correct Option is A
Approach Solution - 1
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:
\( \chi = \frac{\mu}{\mu_0} - 1 \)
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