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What all the maths is about in Maxwell's equations! Click on the green arrow to return to the previous page
What came out of Maxwell's equations was the notion that electric fields and magnetic fields are real, tangible entities that exist and that electric fields can create magnetic fields which in turn can create electric fields again and so on. Most people are familiar with the magnetic field surrounding bar magnets and can feel the attraction when opposite poles are brought together or the repulsion when like poles are moved towards one another. These are real and the magnetic lines of force can be seen using using iron fillings. From this we see that the lines of force are closed loops going into and out from the ends of the bar magnet. We see that the lines spread out in three dimensions from the ends of the magnet diverging outwards into the space surrounding them.
Suppose we tied a piece of string around the bar magnet and swung it like a pendulum back and forth and again try to imagine the changing magnetic field. The magnetic field is no longer static but dynamic varying in both position at any point we choose along the path it swings in and also in time. To measure the effects felt at any point in space we choose, we need mathematical notation to describe the field properties. We need to account for the strength of the magnetic field at our point and the direction it is moving in. Mathematical quantities called vectors do this as they are defined by their size and the direction they point in. If we further imagine our point in space to lie on a plane or surface through which the magnetic field is passing with each swing of the bar magnet pendulum, we would need maths to describe how the magnetic field interacted with the surface or how it flowed through the surface. Did the lines of force diverge as they moved into the surface with one swing and then did they reverse on the return swing? There is a mathematical quantity called divergence which can measure this.
If now we start swinging the magnet in a circular motion faster and faster, the magnetic field is changing rapidly through any point we choose in space and we can begin to visualize swirling patterns of magnetic force as the field around the bar magnet rotates. Maxwell coined a term he called "Curl" to define this and a good way to visualise this is to use the analogy of a tornado to represent the rotating magnetic field. In the image below the direction the tornado pulls things up from the ground is the vector "w" and the curl of the tornado is a special product of this with the gradient. This gradient operator in vector calculus is given the symbol of an upside down triangle and is called "del". Calculus is intimately associated with gradients and how quantities change as you consider smaller and smaller intervals up to a limiting value.The curl has a positive value in the region of space bounded by the tornado whereas the curl at a point distant from the tornado has zero value. We all know how damaging tornados are and so another way to extend the analogy is to consider the rotation as the angular momentum of the contents of a given region of space. All the arguments put forward and mathematics apply equally to electric fields surrounding moving charges like electrons accelerating or decelerating in space or moving along a conductor. The motions of these fields are dynamic and varying in time and quantities which vary in time are analysed more easily using calculus so to account for all the properties of the field at any point we need vector calculus which has its own rules and operators in the same way arithmetic uses plus and minus, times and divide. Its just that the operators in vector calculus are less familiar to most people.