Exercise 12: Electric Potentials and Fields

python homework

Abstract

In this exercise we discuss the electric potential of a hollow metallic prism with a solid, metallic inner conductor. The inner conductor is held at $V=1$ and the walls of the prism at $V=0$.

Introduction and Background

Classical mechanics explores concepts such as force, energy, potential etc. Force and potential energy are directly related. A net force acting on any object will cause it to accelerate. As an object moves in the direction in which the force accelerates it, its potential energy decreases: the gravitational potential energy of a cannonball at the top of a hill is greater than at the base of the hill. As it rolls downhill its potential energy decreases, being translated to motion, inertial (kinetic) energy.
It is possible to define the potential of certain force fields so that the potential energy of an object in that field depends only on the position of the object with respect to the field. Two such force fields are the gravitational field and an electric field (in the absence of time-varying magnetic fields). Such fields must affect objects due to the intrinsic properties of the object (e.g., mass or charge) and the position of the object.
Objects may possess a property known as electric charge and an electric field exerts a force on charged objects. If the charged object has a positive charge the force will be in the direction of the electric field vector at that point while if the charge is negative the force will be in the opposite direction. The magnitude of the force is given by the quantity of the charge multiplied by the magnitude of the electric field vector.

The electric potential created by a charge Q is V=Q/(4πεor). Different values of Q will make different values of electric potential V (shown in the image).

Content

• Solve for the potential in the prism geometry in Figure 5.4
  Figure 5.4
  

• Numerical approach
  In regions of space that do not contain any electric charges, the electric potential obeys Laplace's equation (2 dimension).
  

  $$\frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\partial y^2}=0$$
  Use Jacobi method we have the solution:
  $$V(i,j)=\frac{1}{4}[V(i+1,j)+V(i-1,j)+V(i,j+1)+V(i,j-1)]$$
  In the program, we use:
  $$V_{n+1}(i,j)=\frac{1}{4}[V_n(i+1,j)+V_n(i-1,j)+V_n(i,j+1)+V_n(i,j-1)]$$
  in each step.

• Code and Program

  1. Firstly, we consider the electric field of a plate electrode
     Click here to get python code1.
     Figure 1: Equipotential lines
     
     Figure 2: 3D model
     
  2. The electric potential of a hollow metallic prism with a solid, metallic inner conductor.
     Click here to get python code2.
     Figure 3: Equipotential lines
     
     Figure 4: 3D model
     
     Figure 5: Electric field
     

Thanks to

Zhongzhi Guo
Xiao Guo