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What is general strategy to find the resistance of medium?
These are some of the examples

  1. Two metal balls of the same radius $a$ are located in a homogeneous poorly conducting medium with resistivity $\rho$. Find the resistance of the medium between the balls provided that the separation between them is much greater than the radius of the ball.
  2. A metal ball of radius a is located at a distance $l$ from an infinite ideally conducting plane. The space around the ball is filled with a homogeneous poorly conducting medium with resistivity $\rho$. In the case of $a <<l$ find: (a) the current density at the conducting plane as a function of distance r from the ball if the potential difference between the ball and the plane is equal to V; (b) the electric resistance of the medium between the ball and the plane.

Now here we can't apply $$R=\rho \frac{l}{A}$$ As we dont have finite area. In first one We can get $V$ but still we can't find $R$ by Ohm's law beacause $I$ is not known. Any idea how to solve these ? (I don't expect a solution to the problems above they are just to explain my question )

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A general solution is to

  1. Assume a potential difference between the electrodes (the two balls in the first problem, or the ball and the plane in the second problem).

  2. Calculate the electric field present in the medium surrounding the electrodes.

  3. Use the microscopic form of Ohm's Law:

    $$\vec{J}=\sigma\vec{E}$$

    and integrate over a closed surface surrounding one or the other of the electrodes to find the net current between the balls.

If there's no clever way to calculate the field or the integral, you might need to resort to numerical methods such as finite elements rather than analytical methods.

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