Soil Resistivity

Abstract

Soil is defined as the top layer of the earth’s crust. It covers most of the land on the earth. It is made up of minerals (rock, sand, clay and silt), air, water and organic materials. The organic materials are formed from the dead plants and animals. There are many properties of soil. Although varying soil properties of different kinds can be observed small area, the electric power utility companies are interested in the electrical properties of the soil, especially the specific resistance or resistivity. Soil resistivity is one of the important factors which plays a vital role in the design and analysis of ground resistance. The dry soil with small particles acts as nonconductor of current. Sands, rocks and loams are some examples of nonconductors. The resistivity of soils drops down when the water content in the soil is more. In this chapter, different types and characteristics of soil, and size of earth electrode and different ground resistance measurement methods will be discussed.

Exercise Problems

• 5.1 The expression of electric potential in Cartesian coordinates is $V(x,y,z) = x^{2} y - z^{2} + 8$. Determine the (i) numerical value of the voltage at point $P(1, - 1,2)$, (ii) electric field, and (iii) verify the Laplace equation.

• 5.2 The electric potential in Cartesian coordinates is given by $V(x,y,z) = {\text{e}}^{x} - {\text{e}}^{ - y} + z^{2}$. Determine the (i) numerical value of the voltage at point $P(1,1, - 2)$, (ii) electric field at the point $P(1,1, - 2)$, and (iii) verify the Laplace equation.

• 5.3 The expression of electric potential in cylindrical coordinates is given as $V(\rho ,\phi ,z) = \rho^{2} z\cos \phi$. Determine the (i) numerical value of the voltage at point $P(\rho = - 1,\phi = 45^{ \circ } ,z = 5)$, (ii) electric field at the point $P(\rho = - 1,\phi = 45^{ \circ } ,z = 5)$, and (iii) verify the Laplace equation.

• 5.4 The electric potential in spherical coordinates is given by $V(r,\theta ,\phi ) = 5r^{2} \sin \theta \cos \phi$. Determine the (i) numerical value of the voltage at point $P(r = 1,\theta = 40^{ \circ } ,\phi = 120^{ \circ } )$, (ii) electric field at the point $P(r = 1,\theta = 40^{ \circ } ,\phi = 120^{ \circ } )$, and (iii) verify the Laplace equation.

• 5.5 In Cartesian coordinates, the volume charge density is $\rho_{v} = - 1.6 \times 10^{ - 11} \varepsilon_{0} x\;{\text{C/m}}^{3}$ in the free space. Consider $V = 0$ at $x = 0$ and $V = 4\,{\text{V}}$ at $x = 2\;{\text{m}}$. Determine the electric potential and field at $x = 5\;{\text{m}} .$

• 5.6 The charge density in cylindrical coordinates is $\rho_{v} = \frac{25}{\rho }\,{\text{pC/m}}^{ 3}$. Consider $V = 0$ at $\rho = 2\,{\text{m}}$ and $V = 120\,{\text{V}}$ at $\rho = 5\,{\text{m}}$. Calculate the electric potential and field at $\rho = 6\,{\text{m}}$.

• 5.7 Two concentric spherical shells with radius of $r = 1\,{\text{m}}$ and $r = 2\,{\text{m}}$ contain the potentials of $V = 0$ and $V = 80\,{\text{V}}$ respectively. Find the potential and electric field.

• 5.8 Determine the potential of a rectangular object of infinite length. Consider $a = b = 1\,{\text{m}}$, $V_{0} = 50\,{\text{V}}$, $x = \frac{3a}{2}$ and $y = \frac{b}{2}$.