Calculate the electric field at the origin due to an infinite number of charges as shown in the figure.

Three charged particles $A, B$ and $C$ with charges $-4q, 2q$ and $-2q$ are present on the circumference of a circle of radius $d$. The charged particles $A, C$ and the centre $O$ of the circle form an equilateral triangle as shown in the figure. The electric field at $O$ along the $x$-direction is

Four charges,each of magnitude $q$ coulomb,are placed at points $(-1,0,0), (1,0,0), (0,-1,0)$ and $(0,1,0)$ in the $xy$-plane. The distances along the axes are measured in metres. The magnitude of the electric field at the point $(0,0,1)$ on the $Z$-axis is

Four point charges $-q, +q, +q$ and $-q$ are placed on the $y$-axis at $y = -2d, y = -d, y = +d$ and $y = +2d$,respectively. The magnitude of the electric field $E$ at a point on the $x$-axis at $x = D$,with $D >> d$,will vary as:

An infinite line charge produces a field of $9 \times 10^4 \text{ NC}^{-1}$ at a distance of $2 \text{ cm}$. Calculate the electric field produced at a distance of $3 \text{ cm}$.

The electric field due to a point charge $2q$ at a distance $r$ is $E$. Now,if charge $q$ is uniformly distributed over a thin spherical shell of radius $R$,the electric field at a distance $\frac{r}{2}$ $(r \gg R)$ from the center of the thin spherical shell is $E'=$ . . . . . . .