Why Does Non-Constant Conductivity Affect Electric Field Divergence?

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fricke
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Why is the divergence of electric field not zero for a material with non-constant conductivity?
 
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fricke said:
Why is the divergence of electric field not zero for a material with non-constant conductivity?
There are times when a material with non constant conductivity will have an E field with zero divergence. Can you think when that happens? So what is different?