[2008EM/HMW-04]

IMSc-IV Physics-IV : Electricity and Magnetism Jan-Apr 2008  Set-IV : Tutorial-III Electric Potential

• A conical surface (an empty ice cream cone) carries a uniform surface charge density $\sigma$. The height of the cone is $h$ and the radius of the top is $R$. Find the potential difference between the vertex and the center of the top.
• Find the potential on the axis of a uniformly charged cylindrical shell, a distance $a$ from the center. The length of the cylinder is $L$, its radius $R$ and the surface charge density is $\sigma$.
• The electric field due to an infinite long line charge, glued on the $z$-axis with uniform charge density $\lambda$, can be obtained using Gauss law. Use the answer for electric field derived in the class and integrate $$ \int_P^Q \vec{E}\cdot\overrightarrow{dl}$$ along a suitably selected path to find the potential difference between two points $P$ and $Q$. The coordinates of $P,Q$ are $(L,0,0)$ and $(0,2L,2L)$, respectively.
• Use Gauss law to find the electric field due to two infinitely long coaxial cylindrical shells carrying surface charge densities $\sigma$ and $2\sigma$ and having radii $R$ and $2R$ respectively.
• [$\oslash$] A useful integral $$ \int \frac{dx}{\sqrt{x^2+a^2}} = \log( x+\sqrt{x^2+a^2}).$$ Try proving this result later, after the class is over.