The relative magnetic permeability \( (\mu_{r}) \) and the magnetic susceptibility \( (\chi) \) for a magnetic material are related as
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \mu_{r} = 1+\chi \)
In a magnetic material, \( \vec{B} = \mu_{0}(\vec{H}+\vec{M}) \), and the magnetic susceptibility is defined by \( \vec{M} = \chi\vec{H} \).
\( \vec{B} = \mu_{0}(1+\chi)\vec{H} = \mu_{0}\mu_{r}\vec{H} \)
So the relative permeability is \( \mu_{r} = 1+\chi \), a pure number with no \( \mu_{0} \) in it.
Why the other options are incorrect
Option (1) \( \mu_{r} = \mu_{0}(1+\chi) \) — Incorrect. \( \mu_{0}(1+\chi) \) is the permeability \( \mu \) of the material, not the relative permeability \( \mu_{r} = \mu/\mu_{0} \).
Option (2) \( \mu_{r}+1 = \chi \) — Incorrect. This has the 1 on the wrong side; it would give \( \chi = \mu_{r}+1 \).
Option (3) \( \mu_{r}+1 = \mu_{0}\chi \) — Incorrect. \( \mu_{0} \) should not appear in a relation between two dimensionless quantities.
NCERT (2026-27):
Summary – “… For a linear material \( \mathbf{M} = \chi\,\mathbf{H} \). So that \( \mathbf{B} = \mu\,\mathbf{H} \) and \( \chi \) is called the magnetic susceptibility of the material. The three quantities, \( \chi \), the relative magnetic permeability \( \mu_r \), and the magnetic permeability \( \mu \) are related as follows:
\( \mu = \mu_0\,\mu_r \)
\( \mu_r = 1 + \chi \)”