3
$\begingroup$

The equation for and electric field is given by $E=−∇ψ$ where $\psi$ is the potential, and in this case $ψ=−Q\ln r$ where $Q$ is just some constant. I have found its harmonic conjugate to be $−Qθ+c$ where $c$ is some constant. What does it say about the field? I know that if I calculate the field directly from $E=−∇ψ$, I get $E=Q/r$ pointing radially outwards, but I am not sure how to interpret the harmonic conjugate found (is it right?).

Please help! Thank you.

Added: I can see that $\psi$ is only dependent on $r$ and its h.c. only dependent on $\theta$ means that they are orthogonal, but what else about $E$ can I tell from the h.c. of $\psi$? Perhaps I have made a mistake but should the h.c. have a factor $1\over r$ in the term with the $\theta$?

Update: Thank you to everyone who read this question. I believe I have figured it out now. The h.c. I have obtained is correct. The correct interpretation follows directly from the CR equations! I don't know why I didn't see this earlier...

  • 2
    +1 for your care taken in indiacting the status of the question. You could write this up as an answer, and accept it. And, at this cost, you'll have people help you correct minor things and offer you more intuition.2012-01-31

0 Answers 0