Question

Describe the magnetisation (M), magnetic intensity (H), magnetic susceptibility (χ) and permeability (µ, µ_o and µ_r).  

Answer 1

We know that a circulating electron in an atom has a magnetic moment. In a bulk material, these moments add up vectorially and they can give a net magnetic moment which is non-zero. We define magnetisation M of a sample to be equal to its net magnetic moment per unit volume: 

M = m_net/V 

M is a vector with dimensions L A and is measured in a units of Am^-1. Consider a long solenoid of n turns per unit length and carrying a current I. The magnetic field in the interior of the solenoid was shown to be given by 

B₀ = μ₀ nI 

If the interior of the solenoid is filled with a material with non-zero magnetisation, the field inside the solenoid will be greater than B₀. The net B field in the interior of the solenoid may be expressed as 

B = B₀ + B_m 

where B_m is the field contributed by the material core. It turns out that this additional field B_m is proportional to the magnetisation M of the material and is expressed as 

Bm = μ₀M 

where μ0 is the same constant (permittivity of vacuum) that appears in Biot-Savart law. It is convenient to introduce another vector field H, called the magnetic intensity, which is defined by H = B/µ₀ - M 

where H has the same dimensions as M and is measured in units of Am^-1. Thus, the total magnetic field B is written as 

B = μ₀ (H + M) 

If we partition the contribution to the total magnetic field inside the sample into two parts: one, due to external factors such as the current in the solenoid. This is represented by H. The other is due to the specific nature of the magnetic material, namely M. The latter quantity can be influenced by external factors. This influence is mathematically expressed as 

M = χ H 

where χ, a dimensionless quantity, is appropriately called the magnetic susceptibility. It is a measure of how a magnetic material responds to an external field. lists χ for some elements. It is small and positive for materials, which are called paramagnetic. It is small and negative for materials, which are termed diamagnetic. In the latter case M and H are opposite in direction. we obtain, 

B = μ₀ (1 + χ H) = μ₀ μ_r H = μ H 

where μ_r = 1 + χ, is a dimensionless quantity called the relative magnetic permeability of the substance. It is the analog of the dielectric constant in electrostatics. The magnetic permeability of the substance is m and it has the same dimensions and units as μ₀; 

μ = μ₀ μ_r = μ₀ (1 + χ ) 

The three quantities χ, μ_r and μ are interrelated and only one of them is independent. Given one, the other two may be easily determined.