Physics Friday – Maxwell’s Equations

Every once in a while it is important to take those dusty tomes down off the top shelf, or out of those boxes behind the furnace in the basement, and remind ourselves exactly what makes all this stuff tick.  Periodically I plan to post a little something about these fundamentals – “Physics Friday”.

This Friday the topic is one that is crucial for electronics: Maxwell’s equations.  There are a few different ways of presenting these equations, the most common one is in integral form.  This is a great form for introducing the topic, but there are only a handful of highly symmetric problems that you can attack with this.  The next most common is in differential form using the del operator with either the dot product or cross product.  This is a very useful form, although it only holds for Cartesian coordinates, which is rarely the coordinate system of choice for E&M problems.  Another differential form uses the “div” and “curl” operators, which are not only coordinate system independent, they are also very intuitive.  The “div” operator is just that – a diverging field (one that tends to move outward to infinity) and the “curl” operator is a curling field (think of the “right hand rule“).

div\mathbf{E}=\frac{\rho }{\varepsilon_{0}}

A charge density produces a diverging electric field (Gauss’s Law).

div\mathbf{B}=0

No magnetic monopoles means there is no diverging magnetic field (Gauss’s law for magnetism).

curl\mathbf{E}=-\frac{\partial }{\partial t}\mathbf{B}

A time-varying magnetic field generates a curling electric field that tends to oppose it (Faraday’s Law).

curl\mathbf{B}=\mu _{0}\mathbf{J}+\mu _{0}\varepsilon _{0}\frac{\partial }{\partial t}\mathbf{E}

A curling magnetic field is generated by either a current density or a time-varying electric field (Ampere’s Law).

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