I have some medium with a resistivity which depends on position. In this material are two electrodes, is there an integral or something which i could numerically integrate to find the resistance between the electrodes?
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1$\begingroup$ Possible duplicate of Calculating the resistance of a 3D shape between two points $\endgroup$– FlorisCommented Feb 6, 2016 at 16:55
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It depends on the shape of the medium, but basically you could have an infinitesimal resistance, $\mathrm{d}R$, and integrate that. For example, for a uniform area wire of area $A$ and length $\ell$ in which the resistivity varied sinusoidally about some nominal value, $$\rho(x)=\rho_0(1+0.05\sin\left(\frac{5}{\ell}x\right)),$$ then $$R = \int_0^{\ell}\frac{\rho(x)\mathrm{d}x}{A}=\int_0^{\ell}\frac{\rho_0(1+0.05\sin(5x/\ell))\mathrm{d}x}{A}=\frac{\rho_0\ell}{A}+\frac{0.01\rho_0\ell}{A}\left(1-\cos(5)\right).$$
If you medium isn't a wire, the form of $\mathrm{d}R$ will change and will depend on the basic form of Ohm's Law, $$\vec{J}=\frac{\vec{E}}{\rho}.$$
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$\begingroup$ Its not a wire, its a non-homogenous mixture of liquids in a tank, with two electrodes lowered into it. Is there an expression to use for this? $\endgroup$– J. DoeCommented Feb 5, 2016 at 19:49
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2$\begingroup$ That makes it very complicated because you probably don't have functional form for $\rho$ and it would be 2- or 3-dimensionally dependent. Model it as a mesh of small resistors using finite element analysis. $\endgroup$– Bill NCommented Feb 5, 2016 at 19:53
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$\begingroup$ My expression would be "use a multimeter and measure it." Seems ithe resistance would be very temperature-dependent. $\endgroup$– Bill NCommented Feb 5, 2016 at 19:54