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For a homogeneous material characterized by a resistivity $\rho$ (in $\Omega m$) the resistance between any two points of contact is unbounded. Such "infinite" resistance even applies if one point of contact is replaced by a spherical contact area centered around the point. Just check for yourself and calculate the resistance for this latter configuration by integrating $ \rho /(4 \pi r^2)$ from zero to any finite radial distance.

Another way of recognizing this divergence is by dimensional analysis. To get from a resistivity $\rho$ measured in $\Omega m$ to a resistance $R$ measured in $\Omega$ is by dividing $\rho$ by a length scale. This length scale can not be the distance between the contacts, as this would lead to the unphysical behavior of the resistance between two points decreasing with increasing distance. It turns our that the relevant length scale is the linear size $r$ of the electrical contacts: $R \approx \rho / r$.

Physically what is happening is that the electrical field strength diverges towards a current injection point. You have to assume finite contact areas to obtain a meaningful answer.

For a homogeneous material characterized by a resistivity $\rho$ (in $\Omega m$) the resistance between any two points of contact is unbounded. Such "infinite" resistance even applies if one point of contact is replaced by a spherical contact area centered around the point. Just check for yourself and calculate the resistance for this latter configuration by integrating $ \rho /(4 \pi r^2)$ from zero to any finite radial distance.

Another way of recognizing this divergence is by dimensional analysis. To get from a resistivity $\rho$ measured in $\Omega m$ to a resistance $R$ measured in $\Omega$, one has to divide $\rho$ by a length scale. This length scale can not be the distance between the contacts, as this would lead to the unphysical behavior of the resistance between two points decreasing with increasing distance. It turns our that the relevant length scale is the linear size $r$ of the electrical contacts: $R \approx \rho / r$.

Physically what happens is that the electrical field strength diverges towards a current injection point. You have to assume finite contact areas to obtain a meaningful answer.

For a homogeneous material characterized by a conductivity $\sigma$ (in S/m) the resistance between any two points is unbounded. Such "infinite" resistance even applies if one point is replaced by a spherical surface centered around the point. Just check for yourself and calculate the resistance for this latter configuration by integrating $ 1/(4 \pi r^2 \sigma)$ from zero to any finite radial distance.

Physically what is happening is that the electrical field strength diverges towards a current injection point. You have to assume finite electrodes to obtain a meaningful answer.

For a homogeneous material characterized by a resistivity $\rho$ (in $\Omega m$) the resistance between any two points of contact is unbounded. Such "infinite" resistance even applies if one point of contact is replaced by a spherical contact area centered around the point. Just check for yourself and calculate the resistance for this latter configuration by integrating $ \rho /(4 \pi r^2)$ from zero to any finite radial distance.

Another way of recognizing this divergence is by dimensional analysis. To get from a resistivity $\rho$ measured in $\Omega m$ to a resistance $R$ measured in $\Omega$ is by dividing $\rho$ by a length scale. This length scale can not be the distance between the contacts, as this would lead to the unphysical behavior of the resistance between two points decreasing with increasing distance. It turns our that the relevant length scale is the linear size $r$ of the electrical contacts: $R \approx \rho / r$.

Physically what is happening is that the electrical field strength diverges towards a current injection point. You have to assume finite contact areas to obtain a meaningful answer.

For a homgeneous material characterized by a conductivity $\sigma$ (in S/m) the resistance between any two points is unbounded. Such "infinite" resistance even applies if one point is replaced by a spherical surface centered around the point. Just check for yourself and calculate the resistance for this latter configuration by integrating $ 1/(4 \pi r^2 \sigma)$ from zero to any finite radial distance.

Physically what is happening is that the electrical field strength diverges towards a current injection point. You have to assume finite electrodes to obtain a meaningful answer.

For a homogeneous material characterized by a conductivity $\sigma$ (in S/m) the resistance between any two points is unbounded. Such "infinite" resistance even applies if one point is replaced by a spherical surface centered around the point. Just check for yourself and calculate the resistance for this latter configuration by integrating $ 1/(4 \pi r^2 \sigma)$ from zero to any finite radial distance.

Physically what is happening is that the electrical field strength diverges towards a current injection point. You have to assume finite electrodes to obtain a meaningful answer.