Maxwell's Original Equations

Quaternion Formulation

In Maxwell's original quaternion formulation of his equations, the 20 variables are divided into scalar and vector components of the various fields and potentials.

Modern Formulation

Electric Field Vector

E: The electric field vector, which has three components in a 3D space (E_x, E_y, E_z).

The electric field (E) at any point in space represents the force that a positive test charge would experience if it were placed at that point, divided by the magnitude of the charge itself. It's a vector quantity, meaning it has both a magnitude and a direction.

Components in 3D space:

In a three-dimensional Cartesian coordinate system, the electric field vector E can be decomposed into its three orthogonal components:

Ex: Component along the x-axis

Ey: Component along the y-axis

Ez: Component along the z-axis

Mathematically, the electric field vector is expressed as:

Where:

i is the unit vector along the x-axis

j is the unit vector along the y-axis

k is the unit vector along the z-axis

​Visual Representation and Examples

1. Uniform Electric Field:

Imagine you have two parallel metal plates, one positively charged and the other negatively charged. If you place them close enough, there will be a nearly uniform electric field between them, pointing directly from the positive plate to the negative plate. In such a scenario, if the plates are aligned along the x-axis, the electric field might look like:

You can visualize this as a series of straight, evenly spaced arrows pointing in the x-direction between the plates. The length of each arrow represents the magnitude of the electric field, and the direction of the arrow indicates the direction of the force a positive test charge would feel.

2. Radially Outward Electric Field from a Point Charge:

Consider a single positive charge located at the origin of your 3D coordinate system. The electric field produced by this charge will radiate outward in all directions. At any given point in space, the electric field will point directly away from the charge.

In this case, the components of the electric field at a position (x,y,z) would vary depending on the relative location with respect to the charge. You can visualize the field lines as emanating outwards from the charge in every direction.

3. Vortex-Like Electric Field:

Imagine an electric field that swirls around the z-axis, like water going down a drain (except the field lines go on infinitely). Such a field might be represented as:

If you were to draw the field at various points in the x-y plane, you'd see arrows circling around the z-axis, their direction and magnitude depending on their position.

Conclusion:

The three components of the electric field in 3D space offer a comprehensive understanding of how the field behaves at any given point. By studying these components, one can determine not only the strength of the field but also the specific orientation at that location. The beauty of breaking the field into its components is that it allows for a detailed analysis in complex scenarios where the field might not be uniform or might be affected by multiple sources.

Magnetic Field Vector

B: The magnetic field vector, also with three components in a 3D space (B_x, B_y, B_z).

The magnetic field (B) at any point in space represents the magnetic influence at that specific point. It is a vector quantity, which means it has both magnitude and direction. Magnetic fields arise from moving charges (electric currents) and also from the intrinsic magnetism of certain materials and particles like electrons.

Components in 3D space:

In a three-dimensional Cartesian coordinate system, the magnetic field vector B can be broken down into its three orthogonal components:

Bx: Component along the x-axis

By: Component along the y-axis

Bz: Component along the z-axis

Mathematically, the magnetic field vector can be expressed as:

Where:

i is the unit vector in the x-axis direction

j is the unit vector in the y-axis direction

k is the unit vector in the z-axis direction

Visual Representation and Examples

1. Magnetic Field Around a Straight Current-Carrying Wire:

Imagine a long wire carrying an electric current running vertically. The magnetic field produced by this current will circle around the wire. The direction of the field is given by the right-hand rule: if you grip the wire with your right hand with the thumb pointing in the direction of the current, your fingers will curl in the direction of the magnetic field.

In this scenario, if the wire is aligned along the z-axis, the magnetic field will have non-zero Bx and By components (depending on the position relative to the wire) and will form concentric circles around the wire in the x-y plane.

Visualize it as if you dropped a pebble into a still pond. The ripples, or circles, emanate outwards. In this case, the wire is the pebble, and the magnetic field are the ripples.

2. Magnetic Field Due to a Loop of Current:

Now, consider a circular loop of wire lying in the x-y plane carrying an electric current. The magnetic field inside this loop will point upwards (or downwards depending on the direction of the current) along the z-axis.

If you could see the magnetic field lines, they would form a pattern similar to the lines of latitude on a globe, converging at the poles (above and below the loop). At the center of the loop, the field would be strongest and pointing straight up or down.

3. Magnetic Field of a Bar Magnet:

Envision a traditional bar magnet with its North and South poles. Magnetic field lines emerge from the North pole and curve around to enter the South pole. Near the North pole, the magnetic field might be largely in the positive z-direction, while near the South pole, it would be in the negative z-direction. Around the sides, the field would have components in the x and y directions.

You can picture this scenario like the Earth's magnetic field. Visualize the Earth with field lines emerging from the North magnetic pole and entering the South magnetic pole, similar to how we illustrate the Earth's magnetic field in geography textbooks.

Conclusion:

The magnetic field vector's three components in 3D space offer a comprehensive perspective on the field's behavior at any point in space. By examining these components, we can gauge both the field's intensity and specific orientation. This segmented view provides an avenue for a granular analysis in multifaceted scenarios where the field might be influenced by multiple sources or factors.

Electric Charge Density (ρ)

ρ: Electric charge density scalar, which represents the amount of electric charge per unit volume.

Electric charge density ρ, is a measure of how much electric charge is concentrated within a given volume. It's a scalar quantity (meaning it only has magnitude and no direction), and its standard unit in the International System of Units (SI) is coulombs per cubic meter (C/m^3).

Mathematically, it's defined as:

Where:

Q is the total electric charge contained within a volume V.

Visual Representation and Examples:

1. Homogeneous Distribution of Charge in a Volume:

Imagine a transparent gelatin dessert cube on a plate. Now, visualize that tiny glowing particles (representing electric charges) are uniformly distributed throughout the entire volume of this gelatin. Here, every small portion of the gelatin has an equal number of glowing particles. This uniformity implies a constant charge density throughout the volume.

2. Non-Homogeneous Distribution:

Now, let's take that same gelatin dessert cube. Instead of a uniform distribution, imagine most of the glowing particles are concentrated at the center, and as you move outwards towards the edges, the number of glowing particles decreases. In this scenario, the electric charge density ρ, is higher at the center and decreases towards the edges.

3. Surface vs. Volume Charge Density:

Imagine a hollow metal sphere, like a Christmas ornament. If the electric charge is spread only on its surface, we talk about surface charge density, usually denoted by σ (measured in C/m^2). The inside, being hollow, has no charge and hence no volume charge density. Picture the outer surface of the sphere sparkling or glowing, while the inside remains dark.

On the other hand, if the sphere were solid and the charge was uniformly distributed throughout its volume, then it would glow uniformly from every point when viewed from the outside.

4. Charge Density in Everyday Objects:

Consider a regular AA battery. These batteries have positive and negative terminals. If you were to theoretically slice the battery open (note: don't do this, as it's hazardous), the distribution of charge in the electrolyte and the other materials inside determines the electric charge density. You can visualize the charge concentration as regions of varying brightness within the battery's contents.

Conclusion:

Understanding electric charge density provides insight into how charges are distributed in various materials and devices. It's an essential concept in electromagnetism, and its varying distributions can significantly influence the behavior of electric fields and forces within and around a material. By envisioning charge density as the concentration of glowing particles or brightness, we can gain a more intuitive grasp of this crucial concept in physics.

Electric Current Density (J)

J: The electric current density vector, which has three components in a 3D space (J_x, J_y, J_z).

Electric current density J, provides a measure of how much electric current flows through a specific cross-sectional area of a material. It is a vector quantity, meaning it has both magnitude and direction. The direction of J is the direction in which positive charges are moving. Its standard unit in the SI system is amperes per square meter (A/m^2).

Mathematically, it's defined as:

Where:

I is the electric current flowing through a cross-sectional area A.

The components Jx,Jy,Jz give the current density's direction and magnitude in the x, y, and z directions, respectively.

Visual Representation and Examples:

1. River Analogy:

Imagine a river flowing through a landscape. The river's width and depth can vary, as can the speed of the water. The current density is akin to the water's flow speed at a particular point.

Uniform Flow: If the river flows evenly, it's like a conductor with a uniform current density across its cross-section.

Varied Flow: In areas where the river narrows, the water flows faster. Similarly, in a conductor with non-uniform material properties, the current density might be higher in certain regions.

2. Highway Traffic:

Think of a multi-lane highway. The cars represent charge carriers (like electrons).

Uniform Traffic: If cars are moving evenly across all lanes at a consistent speed, this represents a uniform current density.

Lane Closure: If one lane is closed, cars from that lane merge into others, increasing the car density (akin to current density) in the adjacent lanes. Similarly, in a conductor, if part of its cross-section doesn't conduct well, the current density increases in the other parts.

3. Light Through a Mesh or Grate:

Imagine shining a flashlight at a mesh or grate. The light passing through the holes represents the current, and the mesh represents a material.

The brighter spots (where light passes through) show regions of higher current density.

Dark spots (blocked by the mesh) show areas of zero current density.

4. Electromagnetic Devices:

In devices like transformers and solenoids, the core's material properties can cause non-uniform current densities. If you painted a picture of this, areas with higher current densities might be illustrated with brighter or more intense colors, while areas with lower densities would be more faded.

Conclusion:

Electric current density is crucial in understanding how charges move through various materials, especially in devices where material properties aren't uniform across their cross-section. Visualizing it as flowing water, highway traffic, or light intensity can offer an intuitive grasp of this essential concept in electromagnetism.

Electric Scalar Potential (Ψ or V)

Ψ: Electric scalar potential, which describes the potential energy per unit charge in an electric field.

The electric scalar potential, often just called electric potential, is a measure of the potential energy a unit positive charge would have due to its position within an electric field. It's a scalar quantity, which means it has magnitude but no direction. The unit of electric potential in the SI system is the volt (V).

Mathematically, it's defined as the work done by an external agent in moving a unit positive charge from a reference point (often infinity) to a specific point inside the field, without accelerating the charge. The equation is given by:

Where:

W is the work done.

qo is a test charge (usually taken as a unit positive charge).

Visual Representation and Examples:

1. Gravitational Analogy with Hills and Valleys:

Imagine a hilly landscape.

Hills represent regions of high potential. It's like placing a ball at the top; it has a lot of potential energy because it can roll down.

Valleys signify areas of low potential. A ball in a valley has less potential energy as it's already at a lower position.

In electric terms, charges tend to move from high potential to low potential regions, similar to how balls roll downhill.

2. Diver on a Diving Board:

Think of a diver standing on a high diving board.

The higher the board, the more potential energy the diver has. Similarly, a charge at a point in space with higher electric potential has more potential energy.

As the diver jumps and falls, the potential energy converts into kinetic energy. In the electric realm, a charge moving from a high to low potential converts its electric potential energy into kinetic energy.

3. Color Maps in Physics Simulations:

In many electric field simulation software, electric potentials are shown using color maps.

Bright Colors (e.g., red or yellow): Indicate regions of high electric potential.

Darker Colors (e.g., blue or green): Represent areas of low potential.

Charges introduced in this simulation will naturally drift from brighter areas to darker areas, visualizing the movement from high to low potential.

4. Water Pressure in a Vertical Pipe:

Imagine a vertical water pipe filled with water.

The water pressure is highest at the bottom due to the weight of the water above. This is analogous to higher electric potential in certain regions due to the influence of charges.

As you move upwards in the pipe, the water pressure decreases. Similarly, the electric potential can decrease as you move further from a positive charge.

Conclusion:

The electric scalar potential provides a way to describe the potential energy landscape of charges in an electric field. Through vivid analogies like hilly terrains, diving boards, color maps, and water pressure in pipes, one can intuitively grasp this foundational concept in electromagnetism.

Vector Potential (A)

A: The vector potential, which has three components (A_x, A_y, A_z) and from which the magnetic field B is derived.

The vector potential A is an intermediary in understanding the behavior of the magnetic field B. In simple terms, while the electric field has its associated scalar potential, the magnetic field has a vector potential. Its significance lies in simplifying problems in electromagnetism, especially in quantum mechanics. The relationship between A and B is given by:

Where:

B is the magnetic field.

∇ x A represents the curl of A.

This equation says that the magnetic field at any point can be derived from the curl (a measure of the vector's rotation) of the vector potential at that point.

Visual Representation and Examples:

1. Water Flow in a River:

Imagine a river with several small whirlpools.

The overall direction in which water flows represents the vector potential A.

The whirlpools or the circular motion of water particles signify the curl of the vector, which is the magnetic field B.

This analogy helps grasp the idea that the vector potential A doesn't directly represent the magnetic field but provides insight into its behavior.

2. Miniature Tornadoes on a Flat Field:

Picture a vast field with multiple miniature tornadoes sporadically forming and disappearing.

The general wind direction across the field represents the vector potential A.

The circular wind motion in the tornadoes signifies the curl of the wind direction, which can be compared to the magnetic field B.

3. Fans on a Dance Floor:

Imagine a dance floor with overhead fans.

The direction each fan blade points to represents the vector potential A.

The circular motion or the spinning of the fan blades signifies the curl or the actual magnetic field B.

When fans (magnetic sources) are turned on, the vector potential defines the direction, but the spinning fans themselves (the curl) produce what we measure as the magnetic field.

4. Graphics in Electromagnetism Simulations:

In electromagnetism simulations:

The vector potential A can be visualized as field lines pointing in specific directions.

The magnetic field B arises as a twisting or rotation of these field lines, visualized by arrows rotating about the lines, signifying the curl of A.

Conclusion:

The vector potential A is a cornerstone in understanding the behavior of the magnetic field B in electromagnetism. Through engaging visualizations like river flows, tornado formations, fan movements, and simulation graphics, the abstract concept of vector potential can be made more tangible.

Electric Displacement Field (D)

D: The electric displacement field vector, with three components (D_x, D_y, D_z).

In the presence of an electric field, free charges move, creating currents and polarizing certain materials. The electric displacement field D is introduced to simplify the relationship between the electric field E and any material response, particularly in dielectrics (insulating materials). It takes into account both the free charges and the polarized charges within a material.

Mathematically, it's often represented by:

Where:

D is the electric displacement field.

ε0 is the vacuum permittivity.

E is the electric field.

P is the polarization vector of the material.

Visual Representation and Examples:

1. Sponge in a Stream:

Imagine holding a sponge in the stream of a tap.

The water directly flowing from the tap represents the free electric field, E.

As the sponge gets wet, it retains some water. This water retained by the sponge represents the polarization, P.

The total water (direct flow + water in sponge) represents the electric displacement field, D.

The sponge analogy helps us understand how a material (sponge) can retain or get polarized by an electric field (water).

2. Crowd at a Concert:

Consider a crowd in a concert where a popular artist points and directs the crowd to move left or right.

The initial movement direction of the crowd due to the artist's instruction is like the electric field, E.

Some people might get excited and start swaying their friends along with them. This extra movement caused by excited fans is similar to the polarization, P.

The overall movement of the crowd (initial + excited fans) represents the electric displacement field, D.

3. Magnets on a Board:

Place magnets on a board and introduce an external magnetic field.

The initial alignment of the magnets due to the external field is akin to the electric field, E.

If the board has certain properties, it might further influence the alignment of these magnets. This is analogous to polarization, P.

The total alignment of the magnets (initial alignment + board's influence) is similar to the electric displacement field, D.

4. Visuals in Computer Simulations:

In electromagnetic simulations:

The electric field E can be visualized as arrows pointing in the direction of the field.

The polarization P is often represented as smaller arrows or color gradients showing the local material response.

The electric displacement field D encompasses both types of arrows, giving a combined and comprehensive view of the effect of the electric field on the material.

Conclusion:

The electric displacement field D elegantly bridges the raw influence of an electric field E and the intrinsic response of a material, the polarization P. Visualizing it through real-world examples, like a sponge in water or a crowd's movement, helps make this abstract concept more relatable and easier to understand.

Magnetic Field Intensity (H)

H: The magnetic field intensity vector, with three components (H_x, H_y, H_z).

The magnetic field intensity H is a measure of the magnetic field produced by free currents and is especially crucial when studying magnetic materials. When a magnetic material is subjected to an external magnetic field, it tends to magnetize in response. H captures this external cause, while the material's response is captured by another parameter, the magnetization M.

Mathematically, the relationship between the magnetic field B, the magnetic field intensity H, and the magnetization M of the material is:

Where:

B is the magnetic flux density or magnetic field.

μ0 is the permeability of free space.

H is the magnetic field intensity.

M is the magnetization of the material.

Visual Representation and Examples:

1. Water Flow and Paddles:

Imagine a stream of water flowing through a narrow channel. Placed within this channel are paddles that can rotate when water flows past them.

The initial flow of the water, before it interacts with the paddles, represents the magnetic field intensity, H.

As water flows, the paddles rotate. This rotation, or the manner in which they align to the flow, can represent the material’s magnetization, M.

The overall motion (water flow + paddle rotation) indicates the total magnetic field, B.

2. Wind and Windmills:

Think of a landscape dotted with windmills. A gust of wind (airflow) makes the windmill blades rotate.

The strength and direction of the wind symbolize the magnetic field intensity, H.

The rotation of the windmill blades in response to the wind represents the magnetization, M of the material.

The combined effect of the wind's force and the windmill's rotation gives a sense of the overall magnetic field, B.

3. Dance Instructor and Students:

Imagine a dance instructor and students. The instructor shows a step, and the students follow, adding their own flair.

The instructor's steps signify the magnetic field intensity, H.

The students’ added flair to the steps represents the material’s magnetization, M.

The entire dance routine, combining the instructor's steps and the students' interpretations, is analogous to the magnetic field, B.

4. Computer Simulations and Vector Fields:

In magnetic simulations:

H can be depicted as arrows representing its strength and direction.

M, the response of the material, may be shown as smaller arrows or as a color gradient.

The total magnetic field B would then encompass both sets of arrows, providing a full picture of the magnetic scenario.

Conclusion:

The magnetic field intensity H provides a measure of the 'cause' in magnetic phenomena, especially as it pertains to how materials respond to external magnetic fields. By visualizing this through tangible real-world scenarios like flowing water and rotating paddles or a dance instructor leading students, the abstract concept becomes more intuitive and relatable.

Free Charge Density (ρf)

ρ_f: Free charge density scalar, or the electric charge density not bound in atoms.

The free charge density scalar ρf represents the electric charge per unit volume that isn't bound within atoms or molecules. In simpler terms, it's the concentration of loose or free charges in a region of space. These free charges can move under the influence of an electric field and contribute to electrical conduction.

The free charge density, ρf, in a given region is mathematically represented as:

Where:

pf is the free charge density (in coulombs per cubic meter)

qfree is the total amount of free charge within a volume V.

V is the volume in which the charge is considered. in cubic meters

Visual Representation and Examples:

1. Crowded Marketplace:

Imagine a bustling marketplace with vendors at stalls (representing bound charges within atoms) and visitors walking around (representing free charges).

Vendors are fixed at their stalls, similar to how bound charges are fixed in atoms.

Visitors can freely move about in the marketplace, analogous to free charges moving in a material.

The density of the visitors represents pf - the more visitors, the higher the free charge density.

2. Marbles in a Box:

Visualize a clear box filled with both glued and loose marbles.

Glued marbles signify bound charges – they're fixed and can't move.

Loose marbles represent free charges. When the box is tilted or shaken, only the loose marbles roll or move.

The concentration of these rolling marbles is akin to pf.

3. Fish in a Pond:

Think of a pond where some fish are tethered to the bottom (representing bound charges), while others swim freely (representing free charges).

Tethered fish remain in place, unable to move extensively, just like bound charges in atoms.

Free-swimming fish move as they please, similar to free charges in a conducting material.

The number of free-swimming fish per unit volume of water represents pf.

4. Dust Particles in a Room:

Imagine a room with particles in the air, some of which are free-floating while others are attached to surfaces or bigger objects.

Attached particles represent bound charges.

Free-floating particles, moving with air currents or when disturbed, represent free charges.

The concentration of these free-floating particles symbolizes pf.

Conclusion:

The free charge density scalar pf. Quantifies the concentration of charges in a space that can freely move and contribute to electrical conduction. Understanding this concept becomes simpler and more tangible when visualized through everyday scenarios like a marketplace with vendors and visitors, marbles in a box, or fish in a pond. These visual analogies help in grasping the difference between bound and free charges and how the latter impacts electrical phenomena.

Free Current Density Vector (jf)

J_f: Free current density vector, which has three components (J_fx, J_fy, J_fz) and represents the density of free flowing electric charge.

Conceptual Definition: The free current density vector jf represents the rate at which free electric charges (like electrons that aren't bound within atoms) flow through a cross-sectional area. Its vector nature indicates the direction of flow in 3D space, with components signifying the flow in the x, y, and z directions, respectively.

Mathematics of Free Current Density:

The free current density in a specific direction is given by:

Where:

jf is the free current density (in amperes per square meter,

Ifree is the free current flowing through the area A.

A is the cross-sectional area through which the charge flows (in square meters).

Visual Representation:

Water Pipe Analogy:

Imagine you have three transparent pipes filled with blue-tinted water, each pipe pointing in the x, y, and z directions. The blue water in the pipe represents free flowing electric charges (like free electrons).

If the water flows faster in the x-direction pipe than in the other two, it means is greater than

If no water is flowing in the y-direction pipe, is zero.

The speed and volume of the water flowing in each pipe give a visual idea of the magnitude of the current density in that direction.

Conclusion:

The free current density vector Jf provides a measure of the rate at which free electric charges flow through a specified area. Just like the rate of water flow in our pipe analogy, it helps to understand the density and directionality of electric charge flow in conductors.

Original Additions

The other ten variables (not typically included in the modern formulation of Maxwell's equations) account for potential magnetic charges and currents.

Magnetic Current Density Vector

J_m: Magnetic current density vector, which has three components (J_mx, J_my, J_mz) and represents the theoretical flow of magnetic monopoles.

The magnetic current density vector, often symbolized as Jm, is a theoretical construct used in certain advanced electromagnetics analyses.

Example:

In our hypothetical universe, let's say we have a thin wire carrying magnetic charges. This “magnetic wire” carries north-pole monopoles moving in the positive x-direction. The density of these moving monopoles creates a magnetic current of, let's say, 3 Am (Amperes-magnetic). If this magnetic current flows through an area of 0.01 m square perpendicular to it, the magnetic current density in the x-direction Jmx would be:

If there is no movement of magnetic charges in the y and z directions, then Jmy Jmz would both be 0 Am/mSquare.

Technical Details:

Components:

Jmx: Magnetic current density in the x-direction.

Jmy: Magnetic current density in the y-direction.

Jmz: Magnetic current density in the z-direction.

Relation to Magnetic Monopoles:

Magnetic monopoles would produce a magnetic current when in motion, much like moving electric charges produce an electric current. The magnetic current density would represent the amount of magnetic current flowing through a unit area perpendicular to the direction of flow.

Utilization:

In some advanced areas of electromagnetic theory, introducing the concept of magnetic current density can simplify analyses by treating certain problems symmetrically between electric and magnetic fields.

Visual Representation

Imagine a parallel universe where, in addition to electric charges, there are entities called magnetic charges. These magnetic charges can either be north-pole monopoles or south-pole monopoles. When these magnetic charges move, they produce a “current” just like moving electric charges produce an electric current.

Magnetic Charge Density Scalar

ρ_m: Magnetic charge density scalar, or the theoretical amount of magnetic charge per unit volume.

The magnetic charge density scalar, often represented as Pm, is a theoretical measure used to quantify the hypothetical concentration of magnetic charges (or magnetic monopoles) in a given volume.

Example:

In our hypothetical scenario, suppose we have a tiny cube with a side length of 1 cm. Within this cube, there are 10 magnetic monopoles. If each monopole carries a magnetic charge of, let's say, 1 mC (milli-Coulomb), the total magnetic charge inside the cube would be 10 mC.

The volume of our cube is:

So, the magnetic charge density Pm within the cube would be:

This means, for every cubic meter of this space, we have a magnetic charge of due to the concentration of magnetic monopoles.

Magnetic Scalar Potential

Ψ_m: Magnetic scalar potential, which describes the potential energy per unit magnetic charge in a magnetic field.

The magnetic scalar potential, often represented as Ψm is a theoretical concept analogous to the electric scalar potential in electromagnetism. Ψm would describe the potential energy per unit magnetic charge in a magnetic field.

Example:

For a visual demonstration, consider a space where the magnetic scalar potential increases linearly as we move along the x-axis.

Let's say:

Where K is a constant determining the rate of increase of the potential.

Here, the magnetic field B in the x-direction due to the gradient of Ψm would be:

Using our function:

This shows that in this hypothetical scenario, there's a constant magnetic field in the negative x-direction throughout the space due to the linear increase of the magnetic scalar potential along the x-axis.

Magnetic Vector Potential

A_m: Magnetic vector potential, which has three components (A_mx, A_my, A_mz) and from which the hypothetical magnetic charge density ρ_m would be derived.

The magnetic vector potential, represented as Am, is a theoretical concept in electromagnetic theory, used analogously to the electric vector potential. Just as A (electric vector potential) is related to the electric current density, Am would be related to the hypothetical flow of magnetic monopoles (or magnetic current density). Am would play a role similar to the electric vector potential A in our current understanding of electromagnetism.

Example:

Let's consider a scenario where our magnetic vector potential is a simple function of spatial coordinates:

If we want to find the electric field E from this, we'd calculate:

Performing this curl operation:

Which yields:

So, in this hypothetical scenario, given the magnetic vector potential Am as described, the resulting electric field at any point in space would be a function of z and y coordinates only.

Technical Details:

Definition: In regions where there are no magnetic monopoles, the magnetic field, B is the curl of the electric vector potential A. The electric field, E, would be the curl of the magnetic vector potential Am.

Relation with Magnetic Field: Given the existence of magnetic monopoles, the relationship between Am and E would be:

Units: Following the symmetry with electric vector potential, the units of the magnetic vector potential would likely be volt-seconds per meter. (V·s/m) or similar units.

Visual Representation

Imagine a river flowing through a landscape. The speed and direction of the water at any given point represents the magnetic vector potential. Now, imagine placing a paddlewheel in the river. The way this paddlewheel spins, due to the river's flow, represents the electric field that arises from the magnetic vector potential.

Magnetic Displacement Field Vector

D_m: Magnetic displacement field vector, with three components (D_mx, D_my, D_mz).

The magnetic displacement field vector, denoted as Dm, is a theoretical construct that parallels the electric displacement field D. Just as D relates to the electric field and the electric charge density, Dm would relate to a hypothetical magnetic field caused by magnetic charges.

Example:

Suppose we have a magnetic field given by:

And a hypothetical magnetic charge density of:

Using the relationship:

And taking

Using the given value for μ0:

This means that the magnetic displacement field is mostly dominated by the hypothetical magnetic charge density in this case, with a small contribution from the magnetic field itself.

Technical Details:

Definition: The magnetic displacement field Dm would be proportional to the magnetic field B n free space, plus an additional term related to the hypothetical magnetic charge density ρm. Analogously to the electric displacement field, this relationship would be given by: where μ0 is the permeability of free space.

Relation with Magnetic Field: Given the existence of magnetic monopoles, the magnetic field B can be expressed as:

Units: Following the symmetry with the electric displacement field, the units of the magnetic displacement field would be analogous and can be represented as Weber per square meter (Wb/m^2).

Visual Representation:

Imagine you're in a forest with trees of varying densities. The magnetic field B is like the actual number of trees you see in any patch of land. Now, suppose there are certain invisible magical trees that only special beings (hypothetical magnetic monopoles) can interact with. The total trees, including these magical ones, represent the magnetic displacement field Dm. It encapsulates both the regular trees and the presence of these special trees.

Magnetic Field Intensity for Magnetic Charges Vector

H_m: Magnetic field intensity for magnetic charges vector, with three components (H_mx, H_my, H_mz).

The Hm vector, or the magnetic field intensity for magnetic charges, is a theoretical construct, mirroring the H (magnetic field intensity) in the realm of the hypothetical magnetic charges. In essence, while H provides a measure of the magnetic field produced by electric currents, Hm would describe the field produced by the movement of magnetic charges.

Example:

Suppose we have a magnetic field given by:

And a hypothetical magnetic current density vector:

Using the relationship:

And taking

After calculation, you'll get an Hm vector with very large values due to the division by the small value of μo , minus the contributions of the magnetic current density vector.

Technical Details:

Definition: Given the existence of magnetic monopoles, Hm would be defined analogously to H in relation to the magnetic field B and the hypothetical magnetic current density Jm. This relationship would be expressed as: where μo is the permeability of free space.

Relation with Magnetic Field: Just as H relates to the electric currents that generate a magnetic field, Hm would be related to hypothetical magnetic currents generating a magnetic field.

Units: The units of Hm would be A/m (Amperes per meter), representing the hypothetical magnetic current per unit length.

Visual Representation

Imagine a river with water currents flowing through it, which represents our familiar electric current in the context of magnetic fields. Now, picture another parallel river, but instead of water, it flows with a silvery, magnetic liquid. This is our hypothetical magnetic current. The flow intensity and direction of this magnetic river represent the Hm vector.

ρ_fm: Free magnetic charge density scalar, or the hypothetical amount of free magnetic charge per unit volume.

J_fm: Free magnetic current density vector, which has three components (J_fmx, J_fmy, J_fmz) and represents the hypothetical density of free flowing magnetic charge.

ϕ_e: Electrodynamic potential scalar.

ϕ_m: Magneto-dynamic potential scalar.

The Original Equations

Maxwell's original work in electromagnetism was built upon the experimental results of many scientists before him, including Michael Faraday. Faraday envisioned electric and magnetic fields as lines of force flowing through space, and Maxwell sought to develop a mathematical model of this concept.

Maxwell's equations originally made use of a “sea of molecular vortices” in the luminiferous aether. This was a mechanical view of the aether, but it's the context in which his 20 equations were formulated.

The original 20 equations can be somewhat intimidating because of their number, but also due to the context in which they were formulated. Here is a broad review:

Electric field equations:

Divergence of the electric displacement D

Curl of the electric field E

Definition of D in terms of the electric field E

Relation of free electricity to E

Magnetic field equations:

Divergence of the magnetic field B

Curl of the magnetic field H

Definition of B in terms of the magnetic field H

Relation of free magnetism to H

Electromotive force equations:

Line integral of E around a closed loop

Line integral of H around a closed loop

Equations for the electric current:

Divergence of the conduction current

Definition of the conduction current in terms of E

Equations for the displacement current (Maxwell's addition):

Divergence of the displacement current

Definition of the displacement current in terms of rate of change of E

Equations for the electromotive forces of conduction and polarization:

E in a closed loop in terms of the conduction current

E in a closed loop in terms of the displacement current

Equations for the electromotive forces of magnetization:

H in a closed loop in terms of the magnetic conduction current

H in a closed loop in terms of the magnetic displacement current

Additional equations:

The definition of E in terms of the potential

The definition of H in terms of the magnetic potential

The equations were formulated using quaternions, a type of mathematical structure.

The 20 variables come into play across these equations in various ways, representing fields, potentials, currents, and material properties.

It should be noted that many of these equations, as formulated by Maxwell, aren't commonly used in modern physics in their original form, due to their complexity and the subsequent simplifications made by others like Heaviside and Hertz.

Source: A Treatise on Electricity and Magnetism By James Maxwell.