Exercise 7.3: Capacitance of concentric squares

1. Modify your code to treat a concentric square boundary, i. e., a square centered inside a bigger square. The potential of the outer square is 10V and the inner square is set at 5V. The lineat dimensions of the two squares are $L_1=5$cm and $L_2=25$cm respectively. Choose a convenient grid. Plot the equipotential lines.

2. A system of two conductors with a charge $Q$ and $-Q$ respectively has a capacitnce $C$ which is defined by the formula:
   
   $$C=\frac{Q}{\Delta V},$$
   
   
   where $\Delta V$ is the potential difference between the two conductors. The charge can be determined by the expression:
   
   $$\sigma = \frac{{\bf E \cdot n}}{\epsilon _0} = \frac{E_n}{\epsilon _0}$$
   
   
   where $\sigma$ is the surface charge density, and $E_n$ is the component of the electric field normal to the surface and can be approximated by
   
   $$E_n = -\frac{\Delta V}{\Delta r},$$
   
   
   where $\Delta V$ is the pottential difference between a boundary cell, and the adjacent interior at a distance $\Delta r$. Use the results of the previous part to calculate $\Delta V$ for each point next to the square surfaces. Use this information to obtain an estimate of $E_n$ for the two surfaces, and teh charge per unit of length on each electrode. are the two charges equal and opposite in sign? Calculate the capacitance in Farads.

3. Move the inner square 1cm off center and repeat the calculations of parts 1. and 2. How do the equipotential surfaces change? Is there any qualitative difference if we set the center conductor potential to -5V instead of +5V?