Electrodynamics, the study of the byproducts of interactions of electric current with magnets or with other currents, is much more ubiquitous than its name may suggest. Without the discovery of this field of physics, the appliances we take for granted today, such as our phones, computers, cars, etc., as power generation from nuclear and hydroelectric plants, would not be functional.
Article Outline
An introduction, parallel plate capacitors, and the Lorentz Force equation
Maxwell's Equations (integral form) and how they relate to electromagnetic waves
Divergence and Stokes' Theorems and intuition for curl and divergence in the differential form of Maxwell's equations
Supplementary materials
Section 1
The field of electrodynamics brings about answers to some truly mind-boggling questions. Why is it that if you have one percent more electrons than protons in your body, you would be able to lift a weight equivalent to that of the entire earth? How can you calculate the electric flux out of a sphere as a function or time? Or the net charge of a field comprised trillions of constantly interacting particles? This article will provide answers to these questions.
The structure and properties of all materials is determined by the electrical and quantum mechanical forces interacting inside of them. For example, electrical conductors are characterized by the ability of their electrons to flow, insulators impede this flow due to the large gap between the valence and conduction bands in their atoms (takes a lot of energy to "jump"), the permeability of a material is determined by how easily magnetic flux builds up inside it, etc.
However, there are numerous fascinating nuances that underlie these properties, one of which is illustrated below given these two parallel plate capacitors, which by definition store electrical energy in an electric field.
In the diagram to the right, let a and b be charged particles, E be the electric field vector, and B be the magnetic field vector. Note that the changing magnetic field induces the changing electric field, and vice versa. This will be discussed later. The forces acting upon the charged particles are given by the Lorentz Force equation, given a charge q and a velocity v :
The magnetic force will act on charged particles a and b if and only if the particles are moving. Notice that the cross between v and B represents the cross product, which is a binary operator that takes two vectors and produces another vector that is perpendicular to both of them.
More on the cross product (quantitative description):
The cross product is defined as the area of the parallelogram made by the vectors v and w.
The area of the parallelogram shown to the right is also equal to its base times its height, the latter of which can be represented by the acute angle between vectors v and w. The cross product between these two vectors can also be mathematically represented using unit vectors and determinants, as demonstrated below.
Lastly, imagining that the magnetic field lines are coming out of the screen from the image on the right, and noting that the resultant force, F, must be perpendicular to both of these vectors, and F will thus act analogously to a centripetal force. The force exerted on a charged particle by a magnetic field will cause the particle to move following a circular path.
Section 2
Maxwell's Equations, assembled in 1861 by James Clerk Maxwell, are four partial differential equations that unified electricity and magnetism. Below they are presented in their integral form, and we will walk through them one by one to get an intuitive understanding. The
integral form is useful to calculate the E and B fields over an entire region of space.
Do not worry if you do not yet understand the meaning of these equations and how they relate to electricity and magnetism. For now, all you need to know if that a partial differential (the squiggly d) represents taking the derivative of an expression containing multiple independent variables with respect to only one of them, and that in this context an integral represents the sum of the electric or magnetic charges over a given area.
Gauss' Laws for electricity and magnetism:
This equation measures the net electric field coming out of/entering a closed surface, where each infinitesimal block of area da (shown in diagram below) has electric field lines exiting normal to the surface.
The electric flux through the closed surface is equal to the total charge Q enclosed by the surface over the permittivity of free space, epsilon naught, which is an ideal physical constant representing how much homogenously distributed electrical energy the vacuum can store. In this case you can consider it to be a scaling constant. This law can be applied to electrostatic problems involving objects that possess symmetries, such as spheres and cylinders.
Similarly, Gauss' Law for magnetism states that there is no net magnetic flux coming out of a
closed surface.
There are as many magnetic field vectors coming out of the surface as vectors going in. This is true because there is no such thing as a magnetic monopole; charges will always flow out of the north end of the magnet and into its south end.
Steps to applying Gauss' Law to find the value of an electric field:
Identify the spatial symmetry of the charge distribution (e.g., spherical, cylindrical, planar)
Calculate the electric flux by evaluating the integral above over the Gaussian surface
Calculate the amount of charge enclosed by the Gaussian surface (varies depending on type of surface and if the point you are evaluating the electric field at is inside or outside the charge distribution) -- in the case of a sphere:
4. Isolate E to calculate the electric field of the charge distribution
Example problems:
Q1: Given a uniform electric field of magnitude of E = 100 Newtons/Coulomb (N/C) that exists in space in the X-direction, calculate the flux of this field through a square with an edge length of 10 cm placed in the Y-Z plane.
A1:
The electric flux is the integral over the electric field times the normal vector to the area. The component of the normal vector to the perpendicular direction has a magnitude of itself multiplied by the cosine of the angle between the perpendicular direction and the vector. Since the Y-Z direction is perpendicular to the X-direction, this angle is zero. The integral of the charge over the surface area is the electric charge in Newtons/Coulomb multiplied by the surface area in meters squared.
Q2: Given a charge Q is uniformly distributed throughout an insulating sphere of radius R, find the magnitude of the electric field at a point R/2 from the center.
A2:
Given Gauss' Law for electricity, we can find the electric field by first substituting for the total enclosed charge, Q. Since charge density is equal to total charge over volume.
Solving for Q via the volume of a sphere,
When alternate values of Q and rho are substituted into our electric flux equation,
and by isolating E,
Faraday-Maxwell Equation:
In a nutshell, this equation states that a rotating/spatially-varying electric field produces a time-varying magnetic field with the opposite sign. If not for this phenomenon, the field produced would be increasingly amplified (not nullified by the negative B field) and consequently violate the principle of conservation of energy. A rigorous proof of this equation is given in the 4th section. Due to the coupling of these two fields, electromagnetic waves (and thus light!) were concretely classified.
The left-hand side of the equation is a line integral over the line C that sums all the parts of the electric field along C, or the magnitude of the electric field rotating along the line. The right-hand side, the derivative with respect to time of the magnetic flux, measures how much the magnetic flux changes per unit time. The RHS may seem ambiguous, though, because how do you decide which surface to perform the surface integral over? For example, it is simple to decide which surface to take the integral over if if the surface is in a plane, but what if the surface doesn't neatly lie in a plane?
However, line integrals are path-independent over a surface if that surface is "simply connected", and the closed integral symbol (the circle in the middle of the integral sign) implies a subset of this property. This means that as long as the surface you integrate over is bounded by the same boundary, the final surface integral you will get remains the same.
For example, the more extruded the surface is, although some of the normal vectors are more severely pointed in the positive direction, there are vectors proportionally severely pointed in the negative direction.
For the ambitious reader, this point is proven rigorously in the appendix.
Practice problem:
Q: A circular loop of wire with a radius of 6 cm is in a 1.8 tesla magnetic field. The loop is completely removed from the magnetic field in 0.25 seconds. What is the induced electromagnetic force in the loop?
A:
Induced voltage is equivalent to the electromagnetic field induced by accelerating electric charges, or the negative change in magnetic flux over time, by the Maxwell-Faraday Law. To find magnetic flux, we must calculate the area times the charge of the magnetic field, and then quantify this change over a 0.25 second period to calculate the net change in the magnetic flux over time, or the induced electromagnetic force.
which means,
Ampere-Maxwell Law:
The closed line integral of the magnetic field is equal to the permeability of free space times the electric current, plus the permeability times the permittivity of free space times the change in electric flux over time. The permittivity and permeability of free space are scaling constants (such that both sides of the equation are in Joules/Ampere) which are the ability of a vacuum to store an electric and magnetic field, respectively. The electric current, I, represents the flow of electric charges.
Therefore, a rotating magnetic field (characterized by the infinitesimal circulation of charges) is produced by a combination of current flow and the change of electric flux with respect to time. What is important to note is that free and bound currents contribute to the total current in the Maxwell-Ampere equation, which affect how this equation behaves in different contexts. Free current is comprised of the charges that are not tied to atoms and travel relatively unimpeded, whereas bound current induced by the polarization/imbalance of changes in a magnetic field. When a magnetic magnetic material is placed in a magnetic field, a magnetic current is generated by the movement of electrons in atoms. However, when an electric field is applied to a dielectric material, positive and negative bound charges separate, polarizing the material.
Practice problem:
Q: Calculate the magnetic field between two capacitor plates, given a path with radius r, and the total radius of the capacitor plates, R, as shown below.
A:
The magnitude of the magnetic field along the plates' boundaries is equivalent to their circumference times the constant magnitude of the magnetic field, and since there is no electric current flowing between the plates, the circulation of the magnetic field is only equivalent to the change in the flux of the electric field over time. Since we're only considering the area within a radius of r, we must apply the corresponding scaling ratio. Lastly, these two expressions are set equal, according to the Ampere-Maxwell law, and B is solved for.
Section 3
This section will introduce you to Stokes' theorem and the Divergence theorem and elaborate upon Maxwell's equations by converting them to their differential form.
An introduction to divergence and curl:
Let's begin with divergence, which has already shown up in integral form in taking the change in the flux of an electric field. Divergence is thus a quantitative representation of the rate of change of the net amount of substance, charge in our case, that enters or leaves an object. The image to the right shows that there is a net (constant) change in flux outwards, which can be quantified by multiplying the surface area of the cube by the magnitude of the flux per unit area, over the volume of the cube, which is the outward flow per unit volume.
Mathematically, flux out of a 3-dimensional area can be written as:
R is the 3D region containing x, y, and z, S is the surface bounding the region R, F is the 3D vector field denoting the "flow" of charges, n is the normal vector, and da is the infinitesimal area unit we're integrating over. Note that, in this formal definition, two integrals are used to signify that the flux is quantified in the normal direction out of each 2D surface. More concretely, the above equation shows that as the limit of the volume of the region approaches 0 (i.e., shrinks about an arbitrary point) of the rate at which the charges exits the surface (flux surface integral) divided by the volume of R. Flux over volume is equivalent to the change in charge density, and our goal, divergence!
Using the nabla operator (explained in more depth in the appendix), the divergence of the vector field F can also be written as shown.
Now, on to curl...
Curl can be thought of as the change in the infinitesimal circulation of a vector field, or the average rotation per unit area. For example, imagine stirring some coffee with a spoon. The liquid will behave similarly to the picture shown to the right, and has positive curl, since the counterclockwise direction is used as a convention to keep curl calculations directionally consistent.
This 3D curl can be mathematically written as:
A is a 2D region containing the point (x, y, z) on a plane perpendicular to its normal vector, n, C is the surface bounding region A, F is the 3D vector field we're calculating the curl of, and ds is the infinitesimal boundary unit we're integrating over. By taking the line integral over the closed curve C, we're finding the circulation of F on that bounding curve. We divide by the area of the region normal to the C to find the change in circulation of charges over time, since the rotation about a region is directly proportional the the area of the region. Note that we're also taking the limit as region A shrinks about the point (x, y, z), in order to represent that curl represents the sum of many infinitesimally small measurements.
Using the same definition of curl as in the first section, the curl of F can also be defined as the nabla operator crossed with the vector F.
Stokes' Theorem and the Divergence Theorem:
The Divergence Theorem states that the sum of the divergence of the vector field, F, over the volume enclosed by the surface, S, is equal to the normal component of the vector over the closed surface. Note that in practice we would generally evaluate the LHS with a triple integral, and the RHS with a double integral, but when presenting these and similar equations, it is okay to always use a single integral to denote the integrand.
Given an infinitesimal cube, assume current goes in one of each of its parallel sides and out of the other (e.g., enters through side 1 and exits through side 5) and the x-direction is leftwards, the y-direction upwards, and the z-direction is coming out of your screen. Given the cube is extremely small, assume charge density is roughly the same anywhere on the cube. Now we will calculate flux through each set of parallel sides and sum the fluxes to demonstrate the relationship between the total divergence over a surface's volume and the flux out of its surface.
Since current is entering through side one and exiting through side 2, their fluxes are negative and positive, respectively.
If the distance separating the two parallel sides is small enough, the field at side 2 is about equal to the field at side 1 plus the change in the value of the field between them multiplied by the distance separating them, or,
Combining the two fluxes, the field at side 1 cancels; only the change in the field between the sides remains.
Since the cube is perfectly symmetric, the same procedure can be applied to the other three sides to get,
Stokes' Theorem states that the surface integral of component of the curl of any vector field, F, normal to the surface is equal to the circulation of F around the closed surface.
We can prove this theorem by cutting an arbitrary closed surface in half, and assume that the two new surfaces created have opposite directions of circulation where they were cut. Lastly, the surface must be broken into infinitesimal cubes, which we will first calculate the circulation over. Assume that the positive x direction is right and the positive y direction is up, and the sign of the vector field depends on which direction it is pointing.
Since the value of the vector field in one direction translated parallel to its original position is equal to the translated position plus the change in the vector field with respect to the direction it is traveling in, times the amount it has been displaced,
Substituting these values into the equation above, we will get a final expression in terms of the curl of the vector field F.
Finally, our proofs of the Divergence Theorem and Stokes' Theorem are complete.
Conversion of Maxwell's Equations into differential form:
We will end our discussion by offering a more nuanced understanding of Maxwell's Equations through a few manipulations. Enjoy!
For Gauss' Law, we must first keep in mind the Divergence Theorem, that charge = volume * charge density, and that if the integral on both sides of an equation is the same, the integrands are too.
Similarly, for Gauss' Law for magnetism, if we note that an integral is 0 only if its integrand is also 0 ->
Via the application of Stokes' Theorem, we can perform a similar conversion for the Faraday-Maxwell Equation (left) and the Ampere-Maxwell Law (right).
Section 4
Supplementary materials are listed below.
Proof that the cross product can be represented by the initial two vectors multiplied by the sine of the angle between them: https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/proof-relationship-between-cross-product-and-sin-of-angle
Learn about all things calculus by scrolling through this resource: https://mathinsight.org/index/general
More in-depth lectures on electromagnetism: https://www.feynmanlectures.caltech.edu/II_toc.html
Griffith's Introduction to Electrodynamics textbook (widely used in undergraduate electrodynamics courses): https://hansandcassady.org/David%20J.%20Griffiths-Introduction%20to%20Electrodynamics-Addison-Wesley%20(2012).pdf
Appendix:
To prove that the integral over a surface gives the same value regardless of the surface's shape given a fixed boundary, we will demonstrate path independence.
An integral is only path independent if the vector field describing it, f, is the gradient of an arbitrary scalar field, F.
The inverted triangle (nabla operator) represents the partial derivative of F with respect to each of the variables it contains. This means,
Let's now demonstrate the validity of this property by
taking the integral of f over an surface S, assuming x, y, and z, and also functions of time. To get our desired result, we first must understand the multivariable chain rule, which states,
This is like the regular chain rule, except, since f is a function of multiple variables, you must take a partial derivative with respect to x, y, or z. Now, to the actual proof!
In the equation above, notice that f is equivalent to the dot product of the gradient of F and dr, which can be equivalently written as ((dx/dt) + (dy/dt) + (dz/dt))dt since dr/dt = (dx/dt) + (dy/dt) + (dz/dt), as r can be thought of as the "net change". In the end, the only two points that matter in taking the integral of a conservative vector field are the start and endpoints, or the boundaries. The same applies to taking a surface integral, which is the 2D analog of a line integral.
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