1. Suppose the total charge Q is distributed uniformly over a ring-shaped with radius R. Find the electric potential at distance z above the center. Use: Q=1μC,R=5 cm and ,z=20 cm.
2. The potential energy between two of neutral atoms or molecules is given by Lennard-Jones:
U(r)=4ε((σ/r)12-(σ/r)6 )
whereεis the depth of the potential and σis the distance at which the potential energy is zero.
a. Based on a model the values of the parameters for water are: ε=15.32 kJ/mol and σ=0.302 nm. Plot the potential energy versus distance for water.
b. Calculate the electric force and plot it by using the finite difference method and compare it with the analytical function: F(r)=24ε(2 σ^12/r^13 -σ^6/r^7 )
3. A simple model of a long DNA is a long tiny cylinderwith radius of 1 nm and dielectric constant of κ and linear charge density of λ.
a. Find the electric field at distance r from DNA.
b. Plot electric field versus distance for λ=-2 |e|/nm and κ=80.
c. Find electric potentialby using finite deference method and compare it with analytical solution: V(r)=-λ/(4πε0κ) (ln?(r)-ln?(r0 ))= -λ/(4πε_0 κ) ln(r)+V_0, ( ?V(r)=V(r)-V_0=-2λ/(4πε0 κ) ln(r)) where V_0is the potential at r_0 and it is a constant and can be chosen equal zero.