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The derivation of Maxwell’s Equations ∇.E ∇×B ∇×E ∇.B by Brandon Speedie Our recent feature on the history of electronics covered many prominent contributors to the field. Two names stand out above others; their work is commonly referred to as the ‘second great unification in physics’. D avid Maddison’s History of Electronics series was published in the October, November and December 2023 issues (siliconchip.au/ Series/404). It mentioned hundreds of people who laid the foundations for modern electronics. Englishman Michael Faraday was one of the standouts in that list, with significant contributions to the understanding of electromagnetics. Faraday was born in 1791 to a poor family. He had an early interest in chemistry, but his family lacked the means to formally educate him. Instead, he became self-taught through books and an unbounded curiosity for experimentation. This practical approach continued throughout his career and set the blueprint for his breakthroughs in electromagnetics, despite having no formal training. Faraday was responsible for many notable discoveries, including the concept of shielding (the Faraday Cage), the effect of a magnetic field on the polarisation of light (the Faraday Effect), the electric motor (an early homopolar type, see Fig.1), the Faraday’s coil and ring experiment demonstrated electromagnetic induction. Source: Ri – siliconchip.au/link/abv3 electric generator (an early dynamo, see Fig.2), and the fact that electricity is a force rather than a ‘fluid’ (as was the understanding at the time). He also theorised that this electromagnetic force extended into the space around current-carrying wires, although his colleagues considered that idea too far-fetched. Faraday didn’t live long enough to see his concept accepted by the scientific community. It was an experiment with an iron ring and two coils of wire in 1831 that proved a defining moment for the vocation we now call electrical engineering. By passing a current through one coil, Faraday observed a temporary current flowing in the second coil, despite the lack of a galvanic connection between them. We now refer to this phenomenon as electromagnetic induction, the property behind many common products such as transformers, electric motors, speakers, dynamic microphones, guitar pickups, RFID cards etc. Most notably, this principle is involved in generating the bulk of our electricity. It was a remarkable achievement, later earning Faraday the moniker, “the father of electricity”. James Clerk Maxwell Maxwell was born in 1831 in Scotland. His comfortable upbringing and access to education contrasted with Faraday. Recognising his academic potential, his family sent him to technical academies and University to foster his curiosity about the world around him. Maxwell had long admired Faraday’s work but understood that he was fundamentally a tinkerer with only a basic understanding of mathematics. Maxwell recognised that his own strengths in mathematics were needed to unify Faraday’s experimental results, along with the work of other notable contributors such as Carl Friedrich Gauss and Hans Christian Ørsted. In 1860, Maxwell’s employment moved to King’s College, where he came into regular contact with Faraday. During this period, he published a four-part paper, “On Physical Lines of Force”, using concepts Faraday had Figs.1 & 2: Faraday’s homopolar motor (left) and Faraday’s disc generator (right). 90 Silicon Chip Australia's electronics magazine siliconchip.com.au Fig.3: an application of the cross product. The torque of an axle can be calculated from the cross product of the radius and force vectors. the two vectors together, then multiplying that by the cosine of the angle between them. The cosine is at a maximum if the two vectors point in the same direction and zero if they are orthogonal. If the vectors this is applied to are unit vectors (vectors of length one), the result is simply the cosine of the angle between them. Divergence (∇●) introduced many decades earlier. It contained the four expressions we now know as Maxwell’s equations that tie together electricity, magnetism and light as a single phenomenon: the electromagnetic force. This is called the ‘second great unification in physics’ because Sir Isaac Newton’s trailblazing work with motion and gravity is considered the first. Vector calculus To understand the notation of Maxwell’s equations, a quick primer on vector calculus is in order. Electromagnetism works in three-dimensional space, which can make mathematical representations confusing. We will cover the basics here, using figures to help visualise the equations. The formulas will follow the differential form derived by Oliver Heaviside from Maxwell’s original paper. Combining Del and the dot product is commonly referred to as the divergence operator. When used on a vector field, it returns a scalar field representing its source at any particular point. For example, calculating the divergence on atmospheric wind speed would give a view of pressure differences. Cross product (×) The cross product is a vector operation to calculate the ‘normal’ of two vectors, resulting in a new vector perpendicular to the two input vectors. Curl (∇×) Combining Del and the cross product yields the curl operator. When applied to a vector field, its result is a vector field that shows the rotation or circulation. Returning to the Meteorology example, calculating the curl of wind speeds in the atmosphere will return vorticity, a measure of cyclone or anticyclone rotation. Negative vorticity usually correlates with low pressure and unstable weather (cyclonic rotation), and positive vorticity with high pressure and fine weather (see Figs.4 & 5). #1) Gauss’ law of magnetism ∇●B=0 Maxwell’s first equation is named after German physicist Henrich Gauss. Fig.4 (top): a wind speed plot showing rotational winds off the east coast of Australia and in the southern ocean. Source: BoM, siliconchip. au/link/abv4 Derivative (d/dt) Fig.5 (bottom): calculating the curl of the wind speed yields the vorticity, which more clearly shows the cyclonic rotation off the east coast (blue) and the anticyclone in the southern ocean (red). Negative vorticity (blue) is associated with atmospheric instability, positive (red) usually means fine weather. The same operation can be used on a 3D electric or magnetic field to derive its source. Source: BoM, siliconchip.au/ link/abv5 The derivative operator, d, is shorthand for the Greek letter delta (Δ), which in mathematics refers to a change or difference. ‘t’ refers to time, so d/dt therefore means the change in a parameter over time or more commonly, ‘rate of change’. The symbol ‘∂’ instead of ‘d’ indicates a partial derivative, which is used when differentiating a function of two or more variables. Nabla / Del (∇) Del is the vector differential operator. It is equivalent to the derivative operator above but can be applied to more than one dimension. In our examples, it will be applied to a 3D field. Dot product (●) A dot product is an operation between two vectors that gives a scalar (numeric) result. The result is equivalent to multiplying the magnitudes of siliconchip.com.au A common example is to derive an axle’s torque from its radius and force vectors. The resulting torque vector is orthogonal to both vectors and points in the direction of its angular force (see Fig.3). Australia's electronics magazine November 2024  91 Fig.6: Gauss’ law of magnetism with reference to a permanent magnet. Any field lines exiting ‘north’ wrap around the magnet and enter at the ‘south’ end. The net magnetic field source is zero for any surface cutting through this field (eg, the square), or for the whole magnet in total. Fig.7: Gauss’ law in an air-gapped capacitor (eg, a tuning gang). A voltage source forces a positive charge to build up on the top plate & a negative charge on the bottom plate. An electric field forms between the charged regions. Fig.8: similar to Fig.7 but with a plastic film dielectric, which has a higher permittivity than air. Electric dipoles in the dielectric orientate themselves to cancel some of the electric field strength, increasing the effective capacitance. Here, B is the magnetic field. Simply stated, the sum of all magnetic fields emanating from an interface will always add to zero. This is most obvious when looking at the magnetic field lines surrounding a bar magnet (see Fig.6). Any field lines exiting ‘north’ wrap around the magnet and enter at the ‘south’ end. Considering any isolated area, or the entire magnet as a whole, there is no magnetic field source. #2) Gauss’ law ∇●E=ρ÷ε Also called Gauss’ flux theorem. Here, E is the electric field, ρ is the charge density (the amount of electric charge per volume) and ε is the permittivity of the material or medium (calculated as ε0εr, where ε0 is the vacuum permittivity and εr is the relative permittivity; in a vacuum εr = 1). This law states that electric charge is the source of an electric field. The strength of that field is proportional to the amount of charge and inversely proportional to the permittivity of the supporting material. This phenomenon is most apparent in a capacitor, where an accumulation of negative charge (electrons) builds up on one plate, and a positive charge (protons or holes) on the other (Fig.7). A dielectric between the plates supports the electric field. Its electric dipoles will be orientated opposite to the direction of the electric field and therefore store some of that electric field strength. Film capacitors use a plastic dielectric such as polypropylene or polystyrene, materials which have a relatively low permittivity, meaning they have few electric dipoles to orientate themselves against the field, leaving it mostly intact (Fig.8). In contrast, ceramic capacitors typically use a much higher permittivity dielectric, such as barium titanate, which will orientate many dipoles in response to the applied field and cancel much of the electric field strength (Fig.9). These dipoles provide a higher capacitance per unit area for ceramic capacitors compared to film caps. #3) Faraday’s law of induction ∇ × E = -∂B/∂t Fig.9: this is like Figs.7 & 8 but with a ceramic dielectric. The high permittivity allows many dipoles to cancel a large proportion of the electric field. This arrangement has very high capacitance per area. 92 Silicon Chip Here, E is the electric field and B is the magnetic field, so ∂B/∂t is the change in magnetic field over time. This equation mathematically formalises Faraday’s coil and ring Australia's electronics magazine experiment. It is the notable law of electromagnetic induction, where a time-varying magnetic field induces an orthogonal electric field. The stronger the magnetic field, or the faster its rate of change, the stronger the resulting electric field. This law is most familiar in rotating generators such as hydroelectric, gas, coal and wind-powered electricity production. As the alternator spins, its rotor produces a changing magnetic field for the stator, inducing an electric field that supplies the grid (see Figs.10 & 11). Similarly, the strings on an electric guitar vibrate when plucked. As they oscillate, they cut through the magnetic field produced by the pickups. This changing magnetic field induces a voltage in the pickup windings, which is amplified by a circuit to drive the speaker(s). #4) Ampere’s law ∇ × B = μJ Here, B is the magnetic field, J is the electric current density in amperes per square metre (A/m2) and μ is the magnetic permeability of the material or medium. The original form of Ampere’s law states that the flow of electric current produces an orthogonal magnetic field. The strength of this field is proportional to the current flow and the magnetic permeability of the material (Fig.12). Ampere’s law is the magnetic equivalent of Gauss’ law. We know that electric charge is the source of the electric field but Ampere’s law shows that the movement of electric charge is the source of a magnetic field. This phenomenon is most apparent in an electromagnet, where a wire is wrapped into a coil. As electric current flows, a magnetic field is produced orthogonal to the wire (Fig.13). Suppose a high permeability material such as iron or ferrite is placed in the coil’s core (Fig.14). In that case, magnetic dipoles orientate themselves in the direction of the magnetic field, increasing its strength. Using an iron-based core to increase magnetic field strength is very common in many magnetically-driven devices. For example, silicon steel is widely used in transformers and the field windings of most electric motors or generators. It is also used in hair clippers, where the 50Hz mains siliconchip.com.au Fig.10 (left): Faraday’s law of induction on a simplified three-phase alternator. The permanent magnet rotor spins, providing a changing magnetic field. An electric field is induced in the top coil, as shown by the voltmeter. Fig.11 (right): the same arrangement as Fig.10 but the rotor has rotated 90°, so the top coil sees no change in the magnetic field. The voltmeter shows no deflection. If the rotor continues to spin, the south side of the magnet will soon be near the coil, inducing an electric field with opposite polarity. Through a full 360° rotation, a sinusoidal waveform is generated, ie, AC voltage. waveform is used to induce a changing magnetic field in cutting teeth, providing an oscillatory motion to trim the hair. Ferrite is another common ironbased material widely used in magnetic products. It is favoured for its unique properties as a poor electrical conductor but a good magnetic conductor (high permeability). That is why it is widely used as a former for high-frequency inductors, in permanent magnets for hobby DC motors and as a source of magnetic fields in loudspeakers. This magazine also commonly features AM ‘loopstick’ antennas in its vintage electronics section, which often have an adjustable ferrite core. By rotating the screw, the ferrite can be moved in or out of the coil, providing an inductance adjustment to ‘slug tune’ the receiver. μ is the permeability of the material or medium, ε is the permittivity of the material or medium and E is the electric field (so ∂E/∂t is the change in electric field over time). The additional term includes the property that a time-varying electric field produces an orthogonal magnetic field. Put simply, the strength of the magnetic field is proportional to the permeability and permittivity of the material, as well as the electric field’s strength and rate of change. When considering this relation, together with Faraday’s law of induction, it can be seen that a time-­varying electric field produces a magnetic field Fig.12: an example of Ampere’s law. Current flowing in a wire produces an orthogonal magnetic field. Maxwell’s addition to Ampere’s law Figs.13 & 14: if the length of wire from Fig.12 is coiled, the magnetic fields constructively interfere, producing a stronger field (left). If a high permeability material is used in the core, magnetic dipoles orientate themselves in the direction of the field, increasing the field strength (right). The original form of Ampere’s law only relates electric current to magnetic field strength. Significantly, Maxwell added a term that relates electric and magnetic fields, termed “Maxwell’s addition”: ∇ × B = μ(J + ε∂E/∂t) Here, B is the magnetic field, J is electric current density in amps (A), siliconchip.com.au and a time-varying magnetic field produces an electric field (see Fig.15). It is a remarkable property; as Faraday so eloquently phrased, “nothing is too good to be true if it be consistent with the laws of nature”. A common example is in the transmission of radio waves by an antenna. Alternating current in the antenna produces a time-varying magnetic field around the conductors, which in turn produces a time-varying electric field that continues to propagate in free space. Some distance away, these fields induce a current in a receiving antenna, allowing the wireless transfer of information. Australia's electronics magazine November 2024  93 Ideal Bridge Rectifiers Choose from six Ideal Diode Bridge Rectifier kits to build: siliconchip. com.au/Shop/?article=16043 28mm spade (SC6850, $30) Compatible with KBPC3504 10A continuous (20A peak), 72V Connectors: 6.3mm spade lugs, 18mm tall IC1 package: MSOP-12 (SMD) Mosfets: TK6R9P08QM,RQ (DPAK) 21mm square pin (SC6851, $30) Compatible with PB1004 10A continuous (20A peak), 72V Connectors: solder pins on a 14mm grid (can be bent to a 13mm grid) IC1 package: MSOP-12 Mosfets: TK6R9P08QM,RQ 5mm pitch SIL (SC6852, $30) Compatible with KBL604 10A continuous (20A peak), 72V Connectors: solder pins at 5mm pitch IC1 package: MSOP-12 Mosfets: TK6R9P08QM,RQ mini SOT-23 (SC6853, $25) Width of W02/W04 2A continuous, 40V Connectors: solder pins 5mm apart at either end IC1 package: MSOP-12 Mosfets: SI2318DS-GE3 (SOT-23) D2PAK standalone (SC6854, $35) 20A continuous, 72V Connectors: 5mm screw terminals at each end IC1 package: MSOP-12 Mosfets: IPB057N06NATMA1 (D2PAK) Fig.15: Maxwell’s addition to Ampere’s law models the propagation of an electromagnetic wave. A changing electric field induces an orthogonal magnetic field, which in turn induces an electric field. The wave propagates in a direction normal to both the electric & magnetic fields, at the speed of light. Source: https://tikz.net/files/electromagnetic_wave-001.png This is also how our sun can power the Earth’s biosphere. As tiny atoms such as helium and hydrogen undergo nuclear fusion inside the sun, they emit electromagnetic waves. These waves propagate through free space as time-varying electric & magnetic fields, eventually reaching Earth, where they are used as an energy source by the flora & fauna on this planet. Theory of relativity Years after Maxwell’s publication, a young Albert Einstein expanded these equations in his own papers. Einstein was fascinated by the concept of light as an electromagnetic wave. The significance of this for him was the notion that the speed of the wave depends only on the permittivity and permeability of the medium it travels through and is therefore invariant of the rela- tive speed of the source (Fig.16). This understanding led Einstein to publish his groundbreaking theory of special relativity in 1905, as well as the well-known mass/energy equivalence formula, E = mc2, where E is energy, m is mass and c is the speed of electromagnetic waves (light). This work was further expanded by Einstein’s theory of general relativity in 1915, which included the force of gravitation in addition to the electromagnetic concepts introduced in special relativity. Maxwell’s equations are so central to this theory that they can be derived from Einstein’s general relativity formulas. Einstein paid tribute to Maxwell later in his career when asked whether he “stands on the shoulders of Newton”, to which he replied, “no, on the SC shoulders of Maxwell”. TO-220 standalone (SC6855, $45) 40A continuous, 72V Connectors: 6.3mm spade lugs, 18mm tall IC1 package: DIP-8 Mosfets: TK5R3E08QM,S1X (TO-220) See our article in the December 2023 issue for more details: siliconchip.au/Article/16043 94 Silicon Chip Fig.16: the speed of electromagnetic waves is proportional only to the permittivity and permeability of the material they pass through. In this prism, red light travels at a different speed than blue (because their wavelengths differ), so they are refracted at different angles. This inspired Albert Einstein to derive his groundbreaking theories of relativity. Source: www.vectorstock.com/35129206 Australia's electronics magazine siliconchip.com.au