An exact formula for the effects of resistor geometry on current noise

In 1968, Kuo and Blank studied the effect of resistor geometry on current noise and concluded that the mean-square noise voltage for a rectangular resistor was inversely proportional to the volume of the resistor. Chen and Rhee, in 1977, extended this idea to nonrectangular shapes by conceptually breaking the resistor into many small curvilinear rectangles and using basic circuit theorems to assemble the results. The present paper derives the limiting form of this expression for differential resistor elements and demonstrates that the geometric factor is actually proportional to the volume integral of the fourth power of the electric field.