Advanced Engineering Math II phần 4 pot

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Question What is the significance of the conjugate potential?

Answer. Since ∞ and § are conjugate,

∂∞

∂x

=

∂§

∂y and ∂∞

∂y

= - ∂§

∂x

In other words, Ô∞ and Ô§ are orthogonal. However, since these gradients are

themselves orthogonal to the lines ∞ = const and § = const, we see that the lines § =

constant are at right angle to the equipotentials. Put another way:

The lines §= const are parallel to Ô∞

and are therefore lines of force (showing the direction of the force)

Looking at the example we just did, the lines of force are given by

§(z) = A[Arg(z-c) - Arg(z+c)] = Const

In the homework, you will see that these too are circles (except for the one degenerate

case when the constant is zero), looking something like magnetic lines of force:

(In fact the are the same thing...)

Using the Complex Potential to get the Electrostatic Field

We know that we can recover the electrostatic field by just taking the gradient of ∞:

E = Ô∞

However,

Ô∞ = “∂∞

∂x , ∂∞

∂y ‘ = “∂∞

∂x , -

∂§

∂x ‘

= ∂∞

∂x - i

∂§

∂x In complex notation

= F'(z)

where F is the complex potential. Conclusion

Conservative Vector Fields and Complex Potentials (Not in Kreyzsig)

If E is a conservative field independent of the Ω-coordinate (or in the complex plane)

then

E = F'(z) ,

where F(z) is the associated complex potential.

Example

Find the electric field corresponding to Example (D).