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For my electromagnetic-field class we are asked to calculate resistance between two electrodes shown as below. enter image description here

The part c(i) is just a simple resistance question using the formula R=pl/a. I don't quite get how to do part c(ii). The mark-scheme suggests using superposition to find current density(J=I/A) then linking current-density with electric-field using well-known ohm's law(J=σE) and then using vector form of Kirchhoff's voltage law to find the voltage. (∫ E.dr). Using E=I/4πr^2 and integrating it with respect to r to find the voltage. Finally to find the resistance using simply the ohm's law R=V/I.

The part I am most confused about the question is in how to express J using superposition theorem and the limits of the integral when we use Kirchhoff's voltage law. The solution is shown as below. enter image description here

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  • \$\begingroup\$ The author is assuming that the current spreads out radially from each electrode (out of the positive one, into the negative one), neglecting the influence.of the other electrode. If the current is I, then the current density goes as the reciprocal of the surface of the hemisphere 'radiating' from each electrode. The extremes of integration imply an origin in the middle of the gap \$\endgroup\$
    – Sredni Vashtar
    Commented May 7, 2023 at 17:36

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