$A$ particle of charge $Q$ and mass $m$ travels through a potential difference $V$ from rest. The final momentum of the particle is

Write expressions for electric potential due to a continuous distribution of charges.

Three charges each of value $+q$ are placed at the corners of an isosceles triangle $ABC$ with sides $AB = AC = 2a$. The midpoints of $AB$ and $AC$ are $D$ and $E$ respectively. The work done in taking a charge $Q$ from $D$ to $E$ is (where $\varepsilon_0$ is the permittivity of free space):

$A$ charge $+q$ is fixed at each of the points $x = x_0, x = 3x_0, x = 5x_0, \dots$ up to $\infty$ on the $X$-axis and a charge $-q$ is fixed at each of the points $x = 2x_0, x = 4x_0, x = 6x_0, \dots$ up to $\infty$. Here $x_0$ is a positive constant. Taking the potential at a point due to a charge $Q$ at a distance $r$ from it to be $\frac{Q}{4\pi\varepsilon_0 r}$,find the potential at the origin due to the above system of charges.

$A$ hollow conducting sphere of radius $R$ carries a charge $(+Q)$ on its surface. Find the electric potential at a distance $r = R/3$ from its center.

Consider two concentric conducting spheres of radii $R$ and $2R$ respectively. The inner sphere is given a charge $+Q$. The outer sphere is grounded. The potential at $r = \frac{3R}{2}$ is