[2008EM/HMW-02]

IMSc-IV Physics-IV : Electricity and Magnetism Jan-Apr 2008 MM : 20 Set-III : Tutorial-II Electric Field Using Coulomb’s Law

• Four charges $q,2q,-q,-2q$ are placed at the consecutive corners of a square of side $a$. Find the electric field at a point distance $z$ above the center of the square. 
• The electric field, due to a line segment of length $2L$ carrying a charge $q$, at a distance $d$ above the mid point is given by $$ E = \frac{1}{4\pi\epsilon_0} \, \frac{q}{d \sqrt{d^2 +L^2}}$$ Using the above result find the electric field at a distance $z$ above the center of a square loop ( side $2L$) carrying a uniform line charge $\lambda$. 
• The electric field is to be computed at a point $P$ on the axis of
  • [(i)] a thin cylindrical shell
  • [(ii)] a thin conical shell
  having uniform surface charge density $\sigma$. Dividing the shell by planes perpendicular to the axis, and using the expression for a electric field due to a uniformly charged ring, write expressions for the electric field as an integral. Compute the integral for the cylindrical shell and show that the field due to the cylindrical shell is given by $$ E = \frac{Q}{4\pi\epsilon_0}\frac{1}{(b-a)}\left(\frac{1}{\sqrt{a^2+R^2}}-\frac{1}{\sqrt{b^2+R^2}}\right) $$