Maximum energy product

Key takeaways

Maximum energy product belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Maximum energy product to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Maximum energy product from memory before moving on to harder problems.

Reference excerpt

In magnetics, the maximum energy product is an important figure-of-merit for the magnetic strength of magnetic materials used to make permanent magnets. It is often denoted (BH)max and is typically given in units of either kJ/m3 (kilojoules per cubic meter, in SI electromagnetism) or MGOe (mega-gauss-oersted, in gaussian electromagnetism). 1 MGOe is equivalent to 7.958 kJ/m3. Magnetic materials with larger maximum energy products need less volume of that material to create a given magnetic B field in a given volume of space. During the 20th century, the maximum energy product of commercially available magnetic materials rose from around 1 MGOe (e.g. in KS Steel) to over 50 MGOe (in neodymium magnets). Other important permanent magnet properties include the remanence (Br) and coercivity (Hc); these quantities are also determined from the saturation loop and are related to the maximum energy product, though not directly.

Definition and significance

The maximum energy product is defined based on the magnetic hysteresis saturation loop (B-H curve), in the demagnetizing portion where the B and H fields are in opposition. It is defined as the maximal value of the product of B and H along this curve (actually, the maximum of the negative of the product, −BH, since they have opposing signs):

( B H ) m a x ≡ max ⁡ ( − B ⋅ H ) . {\displaystyle (BH)_{\rm {max}}\equiv \operatorname {max} (-B\cdot H).}

Equivalently, it can be graphically defined as the area of the largest rectangle that can be drawn between the origin and the saturation demagnetization B-H curve (see figure). The significance of (BH)max is that the volume of magnet necessary for any given application tends to be inversely proportional to (BH)max. This is illustrated by considering a simple magnetic circuit containing a permanent magnet of volume Volmag and an air gap of volume Volgap, connected to each other by a magnetic core. Suppose the goal is to reach a certain field strength Bgap in the gap. In such a situation, the total magnetic energy in the gap (volume-integrated magnetic energy density) is directly equal to half the volume-integrated −BH in the magnet:

E g a p = 1 2 μ 0 ( B g a p ) 2 V o l g a p = − 1 2 B m a g H m a g V o l m a g = − E m a g , {\displaystyle E_{\rm {gap}}={\frac {1}{2\mu _{0}}}(B_{\rm {gap}})^{2}{\rm {Vol}}_{\rm {gap}}=-{\frac {1}{2}}B_{\rm {mag}}H_{\rm {mag}}{\rm {Vol}}_{\rm {mag}}=-E_{\rm {mag}},}

thus in order to achieve the desired magnetic field in the gap, the required volume of magnet can be minimized by maximizing −BH in the magnet. By choosing a magnetic material with a high (BH)max, and also choosing the aspect ratio of the magnet so that its −BH is equal to (BH)max, the required volume of magnet to achieve a target flux density in the air gap is minimized. This expression assumes that the permeability in the core that is connecting the magnetic material to the air gap is infinite, so unlike the equation might imply, you cannot get arbitrarily large flux density in the air gap by decreasing the gap distance. A real core will eventually saturate.

References

Illustrations

Maximum energy product: (BH)max can be graphically defined as the area of the largest rectangle that can drawn in the second quadrant of the B-H loop. — (BH)max can be graphically defined as the area of the largest rectangle that can drawn in the second quadrant of the B-H loop.

Worked examples

Example 1 — a first encounter with Maximum energy product

Start with the simplest possible case. Write down what Maximum energy product claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Maximum energy product before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Maximum energy product ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Maximum energy product

In research: Maximum energy product appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Maximum energy product in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Maximum energy product is common in secondary-school and first-year university syllabi. It links to neighbouring topics Magnetic ordering, Magnetism, Magnetostatics, so understanding it makes those chapters shorter.
In everyday life: Look for Maximum energy product outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Maximum energy product in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Maximum energy product means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Maximum energy product out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Maximum energy product in simple terms?

In magnetics, the maximum energy product is an important figure-of-merit for the magnetic strength of magnetic materials used to make permanent magnets. It is often denoted (BH)max and is typically given in units of either kJ/m3 (kilojoules per cubic meter, in SI electromagnetism) or MGOe (mega-gau…

Why does Maximum energy product matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Maximum energy product?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Maximum energy product.