Energy Loss due to Hysteresis

According to Ewing’s theory of molecular magnetism, a magnetic material even in the unmagnetised condition of an indefinitely large number of molecular magnets endowed with definite polarity. When a magnetizing field is applied, the molecular magnets align themselves in the direction of the field.

During this process, work is done by the magnetizing field in turning the molecular magnetize against the mutual attractive forces. This energy required to magnetize a specimen is not completely recovered when the magnetizing field is turned off, since the magnetization does not become zero. The specimen retains some magnetization because some of the molecular magnets remain aligned in the new formation due to the group forces. To tear them out completely, a coercive force in the reverse direction has to be applied. Thus, there is a loss of energy in taking a ferromagnetic material through a cycle of magnetization. This loss of energy is called hysteresis loss and appears in the form of heat.

Consider a magnetic material having n molecular magnets per unit volume. Let m be the magnetic moment of each magnet and θ the angle which its axis makes with the direction of magnetizing field H.

The magnetic moment m of the molecular magnet can be resolved into a component m cos θ in the direction of H and m sin θ perpendicular to H. The component m cos θ alone contributes to the magnetizing field and the component m sin θ has no effect on the magnetization of the specimen.

If M be the intensity of magnetization, then
        M = Σ m cos θ                                    … (1)
Differenting Eq. (1), dM = d (Σ m cos θ) = - Σ m sin θ dθ.                        … (2)
When M increases to M + dM, θ decreasing to θ – dθ.
The work done by the field in decreasing θ by dθ is given by
    dW = Cdθ                                        … (3)
Here, C = torque for unit deflection = μ₀mH sin θ
. :     dW = μ0mH sin θ x (- dθ) = - μ₀ m H sin θ dθ                        … (4)
The work done by the magnetizing field per unit volume of the material for completing a cycle is,
    W = φ μ₀ H dM = φ H μ₀ dM                                … (5)
Now, B = μ₀ (H + M). For ferromagnetic, M >> H. So B = μ₀M
i.e.,     dB = μ₀ dM                                        … (6)
From Eqs. (5) and (6),
    W = φ H dB                                        … (7)
The area of the B – H loop or μ₀ times the area of the M – H loop gives the energy spent per cycle.

When H is in Am-1 and B is in Wb m^-2, the energy is in joules per cycle per m³ of the material.



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