Hysteresis

What is being plotted

• Take an unmagnetised ferromagnetic material in the form of a rod and place it inside a solenoid. Passing a current through the solenoid generates a magnetic field which magnetises the rod.
• Knowing $\chi$ for the material, the magnetization follows from $M = \chi H$, and knowing $H = nI$ the total field follows from

$$B = \mu_0(H + M)$$

• Plotting $B$ against $H$ as the solenoid current is taken through one complete cycle gives the hysteresis loop of Fig. 11.11.
• The behaviour is non-linear and is not retraced, so the curve carries information about the magnetic history of the sample.

The loop, point by point

• Diagram to draw: axes with $H$ horizontal and $B$ vertical, origin $O$. A dotted initial curve from $O$ up to saturation at $a$ in the first quadrant; then $a \to b$ down the $B$-axis, $b \to c$ across into the second quadrant, $c \to d$ down to reverse saturation in the third quadrant, $d \to e \to f$ back up, and $f \to a$ closing the loop. Label $Ob$ as retentivity and $Oc$ as coercivity.
• $O$: the material is in the unmagnetised state.
• $O \to a$: as $H$ is increased, $B$ also increases, but non-linearly (the dotted line). Near $a$ the field reaches its maximum value — the saturation magnetization condition of the rod, representing the complete alignment and merger of the domains. Increasing $H$ further produces no increase in $B$.
• $a \to b$: the process is not reversible. Reducing the current does not retrace the earlier path, because the earlier domain structure is not recovered. When $H = 0$ (current zero, point $b$), $B$ is not zero. This value of $B$ at $H = 0$ is called retentivity or remanence — some domain alignment is retained even with no magnetising field.
• $b \to c$: increasing the current in the reverse direction brings $B$ to zero at a certain reverse value of $H$. This value of $H$ is called coercivity. At this point the domain axes are randomly oriented with respect to each other.
• $c \to d$: increasing the reverse current further increases $B$ in the reverse sense until it again reaches saturation at $d$. Beyond $d$, increasing $H$ does not increase $B$.
• $d \to e$: reducing $H$ from $d$ makes $B$ reduce along the path $de$. At $e$, $H = 0$ again but $B$ is not zero — the domain structure is present, but the direction of magnetisation is reversed.
• $e \to f \to a$: further increase of the current in the forward direction completes the curve $efa$. Reaching $a$ closes one loop.
• The closed curve is the hysteresis loop, and taking the material once round it is one hysteresis cycle.

What the loop tells you

• Retentivity (remanence): the magnetic induction retained in the material when the magnetising field is reduced to zero.
• Coercivity: the value of the reverse magnetising field required to reduce the retained magnetic induction to zero.
• The area enclosed by the $B$–$H$ loop represents the energy lost as heat per unit volume of the material per cycle of magnetisation. The analogy is with the stress–strain hysteresis of Std XI, where the area inside the loop gave the energy dissipated in deforming the material.
• Because the loop area is an energy loss, a material that is to be magnetised and demagnetised many times per second — a transformer core — must have as narrow a loop as possible.

Hard and soft ferromagnetic materials

• Soft ferromagnetic materials can be easily magnetised and demagnetised; hard ones cannot. Their hysteresis loops have quite different shapes: draw the soft-iron loop as a tall narrow loop hugging the $B$-axis, and the steel loop as a broad fat loop.

Standard results

• $H = nI$, $M = \chi H$, $B = \mu_0(H+M)$; $\mu = \dfrac{B}{H}$ at any point of the curve.
• Retentivity is a value of $B$ (at $H = 0$); coercivity is a value of $H$ (at $B = 0$).
• Area of the loop $=$ energy dissipated per unit volume per cycle, in $\text{J m}^{-3}$.
• Transformer core: soft iron, because it has low coercivity and low retentivity.