Hopkinson's Law

Designing devices that rely on magnetic fields, like motors or transformers, presents a significant challenge. While Maxwell's equations offer a complete description of electromagnetism, applying them directly to complex engineering problems can be overwhelmingly difficult. To bridge the gap between fundamental theory and practical design, engineers and physicists developed a powerful shortcut: the magnetic circuit analogy. This model simplifies magnetic systems by treating them like familiar electrical circuits, with Hopkinson's Law serving as its cornerstone—the magnetic equivalent of Ohm's Law.

This article explores the depth and utility of this elegant concept. The first section, "Principles and Mechanisms," will unpack the analogy piece by piece, defining magnetomotive force, magnetic flux, and reluctance, and demonstrating how they combine to analyze series and parallel magnetic paths. We will also examine the limitations of this ideal model, including real-world effects like fringing and saturation. Subsequently, the "Applications and Interdisciplinary Connections" section will showcase how this framework is used to design essential components like inductors and transformers, explain the forces in permanent magnets, and even connect the fields of electromagnetism, mechanics, and acoustics to explain everyday phenomena.

Imagine you want to build an electromagnet. How much current do you need to send through your coil to lift a paperclip? You could, in principle, attack this problem with the full grandeur of Maxwell's equations, calculating the intricate dance of magnetic fields in and around every atom of your iron core. But that's like using rocket science to bake a cake. It's magnificent, but overkill. Engineers and physicists, like all clever people, love a good shortcut. The shortcut here is one of the most elegant analogies in physics: the ​​magnetic circuit​​.

The idea is breathtakingly simple: let's treat the flow of a magnetic field like the flow of electricity. We know how to analyze electrical circuits with beautiful simplicity using Ohm's Law, $V = IR$. What if we could do the same for magnets? It turns out we can. Let’s build the analogy piece by piece.

In an electrical circuit, a battery provides a voltage, or electromotive force (EMF), that "pushes" the current. In our magnetic circuit, the "push" is provided by a coil of wire. This push is called the ​​magnetomotive force (MMF)​​, often denoted by the symbol $\mathcal{F}$. It’s a measure of the total magnetic "effort" available to drive a field. Where does it come from? Ampère's Law, one of the cornerstones of electromagnetism, tells us that a current creates a circulating magnetic field around it. If we wrap a wire into a coil of $N$ turns and pass a current $I$ through it, the efforts of each turn add up. The total MMF is simply:

This force is measured in Ampere-turns. Notice something crucial: the MMF depends only on the coil ($N$) and the current ($I$) you supply. It is the source, the prime mover of our circuit.

Next, what is "flowing"? In the electrical circuit, it's the charge, which we measure as current, $I$. In the magnetic circuit, what flows is the ​​magnetic flux​​, $\Phi$. You can visualize flux as a collection of magnetic field lines. The total flux, measured in Webers (Wb), is the number of these lines passing through a given area. Our goal in designing an electromagnet is to create and guide this flux.

Finally, what "opposes" the flow? In an electrical wire, resistance, $R$, impedes the current. In a magnetic circuit, the opposition is called ​​magnetic reluctance​​, $\mathcal{R}$. It's a measure of how "unwilling" a material is to let magnetic flux pass through it.

So, we have our cast of characters, a perfect mirror of an electrical circuit:

• ​​Voltage ($V$)​​ $\leftrightarrow$ ​​Magnetomotive Force ($\mathcal{F}$)​​ (The Push)
• ​​Current ($I$)​​ $\leftrightarrow$ ​​Magnetic Flux ($\Phi$)​​ (The Flow)
• ​​Resistance ($R$)​​ $\leftrightarrow$ ​​Magnetic Reluctance ($\mathcal{R}$)​​ (The Opposition)

Putting these pieces together gives us the magnetic equivalent of Ohm's Law, a relationship known as ​​Hopkinson's Law​​:

This beautiful, simple equation is the heart of our magnetic circuit model. It says that the amount of flux you get ($\Phi$) is equal to the MMF you apply ($\mathcal{F}$) divided by the reluctance of the path ($\mathcal{R}$).

But what determines a material's reluctance? The formula is wonderfully analogous to electrical resistance. The reluctance of a simple block of material is:

Here, $l$ is the length of the path the flux must travel, and $A$ is the cross-sectional area of the path. Just like with an electrical wire, a longer, thinner path has more reluctance. The key player here is $\mu$, the ​​magnetic permeability​​ of the material. Permeability is a measure of how easily a material can be magnetized. Materials like iron, nickel, and cobalt are ferromagnetic, meaning they have a very high permeability ($\mu \gg \mu_0$, where $\mu_0$ is the permeability of empty space). Air and most other materials have a permeability very close to $\mu_0$.

This is why we build electromagnets with iron cores. Iron is a "flux conductor." Its high permeability means it has a very low reluctance, creating an easy path for the magnetic flux to follow, channeling it and concentrating it where we want it. Air, by contrast, is a "flux insulator" with high reluctance.

The power of the circuit analogy truly blossoms when we start combining different components. How do we know the rules? They come directly from the fundamental physics of the magnetic field.

The most important rule comes from a deep truth of nature, expressed by Maxwell's equation $\nabla \cdot \mathbf{B} = 0$. This equation says that magnetic field lines never begin or end; they always form closed loops. There are no "magnetic charges" or monopoles to act as sources or sinks. The direct consequence is the ​​conservation of flux​​.

Imagine a river of flux flowing through our circuit. At any junction where the path splits, the amount of flux flowing in must equal the total flux flowing out. This is the magnetic equivalent of Kirchhoff's Current Law.

In a ​​series circuit​​, where components are connected end-to-end (like an iron core with a small air gap cut into it), there are no junctions. Therefore, the flux $\Phi$ must be the same in every part of the circuit—the same flux flows through the iron and through the air gap. Just like series resistors, the total reluctance is simply the sum of the individual reluctances:

To find the total flux, we simply calculate the total reluctance and apply Hopkinson's Law: $\Phi = \mathcal{F} / \mathcal{R}_{\text{total}}$.

In a ​​parallel circuit​​, where the flux path splits and rejoins, the flux divides among the branches. The branches with lower reluctance will draw more flux, just as paths of lower resistance draw more current in an electrical circuit. The MMF "drop" ($\Phi \mathcal{R}$) across each parallel branch must be the same, just as the voltage drop is the same across parallel resistors. This allows us to calculate precisely how the flux distributes itself throughout a complex structure.

Let's use our new tools to uncover something remarkable. Consider a toroidal iron core with a mean length $L_i$ and a very narrow air gap of length $L_g$ cut into it. The iron has a high relative permeability, $\mu_r$ (where $\mu = \mu_r \mu_0$), maybe around 4000. The air gap has $\mu_r = 1$.

The reluctances are:

The total MMF from our coil, $\mathcal{F}$, is dropped across these two series components. What fraction of the MMF is used to push the flux across the air gap? This is like asking what fraction of a battery's voltage is dropped across one of its series resistors. The answer is given by a simple "MMF divider" rule:

Let's plug in some numbers. Suppose our iron path is $L_i = 25 \, \text{cm}$ and our air gap is just $L_g = 1 \, \text{mm}$, with $\mu_r = 4000$. The term $\mu_r L_g$ is $4000 \times 1 \, \text{mm} = 4000 \, \text{mm} = 4 \, \text{m}$. The term $L_i$ is $0.25 \, \text{m}$. The fraction of the MMF drop across the gap is roughly $4 / (4 + 0.25) \approx 0.94$.

This is astounding! Over 94% of the coil's entire magnetomotive force is expended just to push the magnetic flux across a 1-millimeter gap. The long path through the iron is, by comparison, effortless. The air gap, despite its tiny size, dominates the entire circuit's behavior. This is not just a curiosity; it's the central principle behind motors, actuators, and recording heads. We use the low-reluctance iron core as a "flux wire" to efficiently deliver the magnetic field to the air gap, where it can interact with the outside world and do useful work.

The magnetic circuit analogy is powerful, but it is a model, an idealization. In the real world, things are a bit messier. Understanding the limits of our model is just as important as understanding the model itself.

​​Fringing Fields​​: Our model assumes the flux jumps neatly across an air gap within the confines of the core's cross-sectional area. In reality, the field lines bulge outwards, "fringing" into the surrounding space. This bulging increases the effective area of the flux path in the gap. Since reluctance is $\mathcal{R}_g = L_g / (\mu_0 A_{eff})$, this fringing effect decreases the gap's reluctance, allowing more flux to flow for a given MMF. We can make our model more accurate by calculating this effective area, showing the model's flexibility.

​​Leakage Flux​​: Unlike electrical current, which is very well-contained by insulating wires, magnetic flux is "leaky." Not all of the flux lines will dutifully follow the iron core. Some will take shortcuts through the surrounding air, "leaking" from one part of the circuit to another without passing through the intended path (like the air gap). This means the flux might not be perfectly constant in a series circuit, as some of it escapes along the way. In high-precision designs, accounting for this leakage flux is critical.

​​Saturation​​: Perhaps the biggest departure from the simple analogy is ​​nonlinearity​​. We assumed permeability $\mu$ was a constant. For ferromagnetic materials, this is only true for weak fields. As the MMF increases, the magnetic field strength $H$ inside the material grows. The material responds by increasing its flux density $B$. But it can't do this forever. At some point, the material ​​saturates​​—nearly all of its internal magnetic domains are aligned, and it can't offer any more assistance. Its effective permeability drops dramatically. This means reluctance is no longer a fixed number; it becomes dependent on the flux flowing through it, $\mathcal{R}(\Phi)$. Hopkinson's Law becomes $\mathcal{F} = \Phi \mathcal{R}(\Phi)$, a nonlinear equation that is much harder to solve. The simple linear model is an excellent first approximation, but for high-performance magnets operating near their limits, we must face this nonlinearity head-on, often with the help of computers.

The journey from Ampère's Law to a saturating, leaky, fringing magnetic circuit reveals the beautiful arc of physics and engineering: we start with a fundamental law, build a simple and elegant model, use it to gain profound intuition, and then systematically refine it to embrace the complexity of the real world.

After our journey through the principles of magnetic circuits, you might be left with the impression that Hopkinson's Law is a clever, but perhaps limited, analogy. A nice trick for textbook problems. Nothing could be further from the truth. This simple framework, treating magnetic flux like a current flowing through a circuit of reluctances, is nothing short of a secret weapon for engineers and physicists. It transforms the beautiful but often unwieldy complexity of Maxwell's field equations into an intuitive, predictive design tool. It is the bridge from abstract theory to the tangible world of inductors, motors, generators, and transformers. Let's explore how this one elegant idea blossoms across a vast landscape of science and technology.

At the heart of modern electronics lies the ability to create and control magnetic fields. The simplest device for this is the inductor, and Hopkinson's Law is its blueprint. Imagine a simple toroidal core—a doughnut of magnetic material. If we wrap it with $N$ turns of wire, what is its inductance, $L$? The answer from our circuit model is breathtakingly simple: $L = N^2 / \mathcal{R}$, where $\mathcal{R}$ is the total reluctance of the core. The inductance, a measure of the coil's ability to store magnetic energy, is governed by the number of turns squared and the "unwillingness" of the core to permit magnetic flux.

This immediately gives us a recipe for design. Need more inductance? Add more turns, or choose a material with a higher permeability $\mu$ to lower its reluctance $\mathcal{R} = l / (\mu A)$. What if our core is not uniform, but pieced together from different materials? The circuit analogy holds beautifully. Just as resistors in series add up, the reluctances of the different sections of the core simply add together to give the total reluctance, allowing us to engineer custom components with precisely tailored magnetic properties.

You might think that an air gap in a magnetic core is a defect, an unwelcome interruption. In reality, it is often a crucial design element. An air gap has a very high reluctance because the permeability of air, $\mu_0$, is thousands of times smaller than that of iron. This high-reluctance gap often dominates the entire magnetic circuit. This is incredibly useful! It means the total reluctance, and thus the behavior of the device, becomes largely determined by the clean, predictable geometry of the gap, rather than the messy, nonlinear properties of the ferromagnetic core. For devices like electromagnetic relays, where a specific magnetic flux density $B_g$ is needed in the gap to actuate a switch, this predictability is paramount. The magnetic circuit model allows an engineer to calculate this gap field with remarkable accuracy, even accounting for subtle, real-world effects like the "fringing" of the field as it bulges outward into the surrounding space.

Our discussion so far has centered on electromagnets, where the magnetomotive force (MMF) comes from a current-carrying coil. But what about permanent magnets? How can our circuit law possibly describe a solid block of magnetized metal? The trick, a rather elegant one, is to model the permanent magnet itself as a source of MMF—a kind of "magnetic battery"—which also possesses its own "internal" reluctance.

This powerful generalization allows us to understand a classic piece of laboratory wisdom. Why, when storing a strong bar magnet, do you place a piece of soft iron, called a "keeper," across its poles? The keeper, being a soft magnetic material, has a very high permeability and therefore a very low reluctance. It provides an easy, low-reluctance path for the magnetic flux to loop back from the north pole to the south pole. In circuit terms, the keeper acts like a wire short-circuiting a battery. It contains the flux almost entirely within the magnet-keeper assembly, minimizing the external magnetic field. This is vital because it's this external field that creates an opposing "demagnetizing field" inside the magnet, which can slowly weaken it over time. The low-reluctance keeper effectively protects the magnet from itself.

And where this gets really interesting is when we ask: what holds the keeper so tightly to the magnet? The answer lies in the energy of the magnetic field. A magnetic field stores energy, and like any good physical system, it prefers to be in a lower energy state. The energy density in an air gap is much higher than in iron for the same amount of flux. The field, like a collection of stretched rubber bands, pulls the faces of the gap together to reduce its volume and thereby minimize its total energy. The beauty of the magnetic circuit model is that it allows us to calculate the flux density $B$ in the gap, which in turn directly tells us the mechanical attractive force, as the force is proportional to $B^2 A / \mu_0$. We have crossed the bridge from pure electromagnetism to the world of tangible mechanical forces.

So far, we have considered simple, single-loop circuits. But what happens when the path for the flux splits? Nature, it turns out, has a beautiful consistency. Just as electric current is conserved at a junction (Kirchhoff's Current Law), so too is magnetic flux. The total flux flowing into a junction must equal the total flux flowing out. And just as the voltage drop is the same across parallel electrical paths, the MMF drop is the same across parallel magnetic paths.

This simple extension to parallel circuits is the key that unlocks one of the most important inventions in modern civilization: the transformer. Consider a magnetic core with three legs. A coil with $N_A$ turns is wound on one outer leg, and a second coil with $N_B$ turns is wound on the other outer leg. When a current flows in the first coil, it creates a flux. This flux travels down the first leg and arrives at a junction, where it splits. Some of the flux travels through the central leg, and the rest travels through the other outer leg, passing through the second coil. This portion of the flux, linking the second coil, is the essence of mutual inductance. Our parallel circuit model allows us to calculate exactly how the flux divides and, therefore, to find the mutual inductance $M$ between the coils. With this simple "Ohm's Law for magnetism," we have captured the fundamental principle of the transformer, the device that makes our global electrical grid possible.

The true power of a physical law is revealed when it connects seemingly disparate phenomena. Let's see what happens when our magnetic circuits become dynamic. Consider an electromagnetic levitation system, where an electromagnet suspends a ferromagnetic object in mid-air. The height of the air gap, $z$, is now a dynamic variable. As the object bobs up and down, the reluctance of the circuit, $\mathcal{R}(z)$, changes with time. Since the magnetic flux is $\Phi = NI/\mathcal{R}(z)$, a changing reluctance causes a changing flux. And as Faraday taught us, a changing magnetic flux through a coil induces an electromotive force (EMF). Thus, the very motion of the levitated object induces a "motional back-EMF" in the electromagnet's coil, a voltage that is a direct function of the object's velocity $\dot{z}$. This is electromechanical transduction in its purest form—a seamless marriage of mechanics and electricity, forming the basis for countless sensors, actuators, and generators.

As a final illustration of this symphony, let us explain a phenomenon you have certainly experienced: the subtle hum or high-pitched whine emanating from the power adapter for your laptop or phone. Where does this sound come from? It is a magnificent chain of physical causality, and Hopkinson's Law is the conductor.

1. Inside the power adapter is an inductor, likely with an air gap, carrying a current that is not perfectly steady but has a small alternating "ripple" on top of it.
2. This ripple in the current $i(t)$ produces a ripple in the magnetic flux density $B(t)$ in the gap.
3. The attractive force between the core faces is proportional to $B(t)^2$. If $B(t)$ contains a sinusoidal ripple, the squaring operation produces an alternating force at twice the ripple frequency!
4. This tiny, alternating mechanical force causes the physical components of the inductor to vibrate.
5. This vibrating component acts like a miniature loudspeaker, pushing and pulling on the surrounding air molecules and creating pressure waves.
6. These pressure waves travel to your ear, which you perceive as an audible hum.

From a simple circuit analogy, we have followed an unbroken chain of logic through electromagnetism, mechanics, and acoustics to explain a common, everyday annoyance. It is a stunning example of how a single, powerful concept can illuminate the deep and beautiful interconnectedness of the physical world.