Poisson’s and Laplace’s Equations

Abstract

In the previous few chapters, the electric field has been determined using either Gauss’ law or Coulomb’s law. In initial condition, charge distribution or electrostatic potential should be known to apply those laws. There are many practical problems where the charge distribution is not known for every place. There is some complex geometry in high voltage engineering equipment, namely insulators, bushing, surge arrestors, etc. In that case, it is difficult to use Gauss’ law to find their electrostatic potential and electric field intensity distributions. The method of images can be used if the conducting bodies have a boundary with simple geometry. Therefore, some differential equations need to be solved to find the voltage and field distribution around the conductor and air interface of the simple and complex geometry of the electrical engineering equipment. In this chapter, Poisson’s equation, Laplace’s equation, uniqueness theorem, and the solution of Laplace’s equation will be discussed.

Exercise Problems

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

• 4.2 The electric potential in Cartesian coordinates is given by $V(x,y,z)={{e}^{x}}-{{e}^{-y}}+{{z}^{2}}$. Determine the (a) numerical value of the voltage at point $P(1,1,-2)$, (b) the electric field atpoint $P(1,1,-2)$, and (c) verify the Laplace’s equation.

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

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

• 4.5 In Cartesian coordinates, the volume charge density is ${{\rho }_{v}}=-1.6\times {{10}^{-11}}{{\varepsilon }_{0}}x\ \text{C/}{{\text{m}}^{3}}$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 $x=5\,\text{m}$.

• 4.6 The charge density in cylindrical coordinates is ${{\rho }_{v}}=\frac{25}{\rho }\ \text{pC/}{{\text{m}}^{\text{3}}}$. 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 $\rho =6\,\text{m}$.

• 4.7 The concentric spherical shells with radii 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.

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