# 1.3: Return to Maxwell’s Equations

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Maxwell’s equations (1.2.1, 1.2.2, 1.2.3, 1.2.4) form a system of differential equations that can be solved for the vector fields \(\vec E\) and \(\vec B\) given the space and time variation of the four source terms ρ_{f}(\(\vec r\),t), \(\vec {J_{f}}\)(\(\vec r\),t), \(\vec P\)(\(\vec r\),t), and \(\vec M\)(\(\vec r\),t). In order to solve Maxwell’s equations for a specific problem it is usually convenient to specify each vector field in terms of components in one of the three major co-ordinate systems: (a) cartesian co-ordinates (x,y,z), Figure (1.3.10); (b) cylindrical polar co-ordinates (r,θ,z), Figure (1.3.10); and (c) spherical polar co-ordinates (ρ,θ,φ), Figure (1.3.10).

It is also necessary to be able to calculate the scalar field generated by the divergence of a vector field in each of the above three co-ordinate systems. In addition, one must be able to calculate the three components of the curl in the above three co-ordinate systems. Vector derivatives are reviewed by M.R. Spiegel, Mathematical Handbook of Formulas and Table s, Schaum’s Outline Series, McGraw-Hill, New York, 1968, chapter 22. It is also worthwhile reading the discussion contained in The Feynman Lectures on Physics, by R.P. Feynman, R.B. Leighton, and M. Sands, Addison-Wesley, Reading, Mass., 1964, Volume II, chapters 2 and 3.

**The following four vector theorems should be read, understood, and committed to memory because they will be used over and over again in the course of solving Maxwell’s equations.**

Consider a volume V bounded by a closed surface S, see Figure (1.3.11). An element of area on the surface S can be specified by the vector d\(\vec S\)** = \(\hat{n}\)**dS where dS is the magnitude of the element of area and **\(\hat{n}\) **is a unit vector directed along the outward normal to the surface at the element dS. Let \(\vec A\)(\(\vec r\),t) be a vector field that in general may depend upon position and upon time. Then Gauss’ Theorem states that

\[ \int \int_{S}(\vec{\mathrm{A}} \cdot \hat{\mathbf{n}}) d S=\int \int \int_{V} \operatorname{div}(\vec{\mathrm{A}}) d \tau \nonumber\]

where d\(\tau\) is an element of volume. The integrations are to be carried out at a fixed time.

## 1.3.4 Stokes’ Theorem.

Consider a surface S bounded by a closed curve C, see Figure (1.3.12). \(\vec A\)(\(\vec r\),t)** **is any vector field that may in general depend upon position and upon time. At

a fixed time calculate the line integral of \(\vec A\) around the curve C; the element of length along the line C is \(d\overrightarrow{\mathrm{L}}\). Then Stokes’ Theorem states that

\[\int_{C} \overrightarrow{\mathrm{A}} \cdot \overrightarrow{d \mathbf{L}}=\iint_{S} \operatorname{curl}(\overrightarrow{\mathrm{A}}) \cdot \hat{\mathbf{n}} d S \nonumber \]

where **\(\hat{n}\) **is a unit vector normal to the surface element dS whose direction is related to the direction of traversal around the curve C by the right hand rule.