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The Maths Behind Maxwell's Equations

A quick overview of the mathematics involved in Maxwell's equations and why I decided to do this Extended Project

t p

on 19 April 2011

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Transcript of The Maths Behind Maxwell's Equations

The Maths Behind Maxwell's Equations What are they? In the simplest terms Maxwell's Equations are 4 equations that describe how the force of electromagnetism works. The 4 Equations were published by Scottish mathematician James Clerk Maxwell in 1862. Why is this important? Electromagnetism is one of the 4 fundamental forces that govern our physical world and with the possible exception of Gravity, is the one we are most familiar with in our every day lives. What makes Maxwell's Equations so special? Of the 4 fundamental forces, at this point in time, we can only explain one in complete detail, Electromagnetism, through the use of Maxwell's Equations. Why am I interested in them? In short, because I like understanding things. The Longer explanation:
I like understanding things and I especially like understanding things that have been a challenge to master
I like expanding my Maths toolset so that when I come across future problems I can perhaps tackle them in a more efficient manner
Most importantly, Maxwell's equations are a great example of fairly abstract Mathematics being used to explain the physical world, exactly the kind of Maths I want to do at University and beyond.
Maxwell's Equations in differential and integral form:
Ah... 35 hours of online MIT lectures later... Let's look at each individual component of the equations. Vectors
In the equations these are the bold lowercase letters
A vector is simply something that has a magnitude and a direction Vector operations:
There are two main Vector operations; dot products and cross products Dot Products:
A dot product is denoted by the notation a . b
An operation that returns a single value that is not a vector that is formed by combining the lengths of the two vectors and the angle between them
Often used to check if vectors are perpendicular to each other ( dot product = 0) Cross Products:
Denoted by the notation a x b
More complicated to calculate than dot products
Returns a vector that is perpendicular to the other two vectors
Also used to find the area of the parallelogram whose sides are the two vectors Vector Fields:
In Maxwell's equations these are the bolded Uppercase letters (E, B etc.)
A mathematical field that describes a vector at every point in the field
The vector is based upon the position of the point in the field Gradients:
A measure of how quickly a function is increasing or decreasing at a given point
For functions of one variable this is computed through differentiation
This is the process of changing the variable by an infinitely small amount and seeing how much the function as a whole changes
For functions with more than one variable, partial differentiation is used
Each variable is changed seperately keeping the rest constant and the effects are combined.
Gradient Fields
A special case of vector fields where the vectors are the gradient of a function that has a single value at every point in the field
This means that the value of the function does not depend on how you got to the point
A closed loop on a hill always returns the same value for the height
A closed loop on a spiral staircase returns a value for height depending on which direction you went Curl
One of the two main mathematical operators in Maxwell's equations is curl
This describes how much a vector field ISN'T a gradient field
The value of curl has both magnitude and direction (i.e. is a vector)
It is equivalent to saying how steep the spiral staircase is and in what direction it goes
Curl = 0 and the field is defined everywhere, then it is a gradient field Divergence
The other key operator is Divergence
This is a measure of how much a field is being added to at a given point
Imagine a tap in a bath with the vector field being the flow of water at each point
At every point other than the tap Divergence is 0 because there is no water being added
At the tap Divergence has a value because a new flow of water is being added to the field Del Notation
Gradient fields, Curl and Divergence can all be simplified by using Del notation
For a function f f = Gradient field
For a Field F . F = Divergence
For a Field F x F = Curl Integrals
Integrals are a way of summing all the tiny values of a function across a certain range to give the total value
Integrals are denoted by the symbol
A single means to sum all the values along the path of a line in space, known as a line integral
Two is where you sum all the values across a surface in space, also known as a surface integral
Three is where you sum up within a volume in space, known as volume integrals
A circle on the integral sign indicates the integral is taken across a closed loop or surface Divergence theorem
Divergence theorem relates surface integrals and volume integrals in a vector field
It says that the amount of stuff coming out of a closed surface is equivalent to the amount of stuff being created within it Stokes' theorem
Stokes' theorem relates line integrals and surface integrals in a vector field
It says that if you go around a closed loop within a vector field, the value of the integral around the loop is equivalent to the sum of all the curls on any surface bounded by the loop Maxwell's Equations describe the E and B fields, which are vector fields describing Electricity and Magnetism respectively The first equation states that the divergence of an electric field is equal to the charge density divided by a physical constant
The second equation states that the divergence of a magnetic field is zero These both become easier to understand when Divergence theorem is applied
The first equation says that the total electric field leaving a volume is proportional to the charge within
The second states that the total magnetic flux through a closed surface is always 0. This essentially says that electric field lines can originate from charges, whilst magnetic field lines do not have a beginning or an end and simply loop The next two equations involve the curl of the electric and magnetic fields
The first equation states that the curl of the electric field depends on the rate of change of the magnetic field
The second states that the curl of the magnetic field depends on the rate of change of the electric field and any current flowing These become easier to understand when Stokes' theorem is applied
The line integral of E around a closed loop is equal to the rate of change of the magnetic field through a surface bounded by this loop
The magnetic field around a wire carrying current will depend on the current in the wire and any changing electrical field These results, that a changing electric field induces a magnetic field and vice versa are the basis of all transformers, motors and generators It is also possible to combine the equations and show that in a vacuum you get the wave equation which describes a self generating electromagnetic wave travelling at the speed of light This is the basis of all telecommunications and much of our understanding of modern science, especially cosmology
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