Exercise - 4

Three-point charges $Q, 2Q$ and $8Q$ are to be placed on a $10 \mathrm{cm}$ long straight line. Find the position where the charges should be placed such that the potential energy of this system is minimum. In this situation, what is the electric field at the position of the charge $Q$ due to the other two charges? 

A particle of charge $-q$ and mass $m$ moves in a circular orbit about a fixed charge $+Q$. Show that $r^3 \alpha T^2$the law, is satisfied, where $r$ is the radius of orbit and $T$ is time period.

The field potential in a certain region of space depends only on the$x-$coordinate as $V=–{\mathrm{ax}}^3+\mathrm{b}$, where $a\text{ and} b$ are constants. Find the distribution of the space charge $\mathrm{\rho }\left(x\right).$

The electric field strength depends only on the $x$ and $y$ coordinates according to the law, $\overrightarrow{E}=\frac{ a\left(x\hat{\mathrm{i}}+y\hat{\mathrm{j}}\right)}{\left(x^2+y^2\right)}$ where $a$ is a constant, $\hat{\mathrm{i}} \text{and} \hat{\mathrm{j}}$ are the unit vectors of the $X$ and $Y$ axes. Find the flux of the vector $E$ through a sphere of radius $R$ with its centre at the origin of coordinates. Using the above result, also calculate the total charge enclosed by the sphere.

A positive charge is distributed in a spherical region with charge density $\mathrm{\rho }= {\mathrm{\rho }}_0 \mathrm{r}$ for $\mathrm{r} \le \mathrm{R}$ (where ${\mathrm{\rho }}_0$ is a positive constant and $\mathrm{r}$ is the distance from centre). Find out electric potential and electric field at following locations.

(a) At a distance $\mathrm{r}$ from centre inside the sphere.

(b) At a distance r from centre outside the sphere.

Two point charges $q$ and $–2q$ are placed at a distance $6 \mathrm{m}$ apart on a horizontal plane ($x–y$ plane). Find the locus of the zero potential points in the $x–y$ plane.

Two metallic balls of radii $R_1$ and $R_2$ are kept in vacuum at a large distance compared to their radii. Find the ratio of the charges on the two balls for which electrostatic energy of the system is minimum. What is the potential difference between the two balls for this ratio? Total charge of the balls is constant. Neglect the interaction energy. (Charge distribution on each ball is uniform)

Consider an equilateral triangle $ABC$ of side $2a$ in the plane of the paper as shown. The centroid of the triangle is $O$. Equal charges $\left(Q\right)$ are fixed at the vertices $A, B$ and $C$. In what follows consider all motion and situations to be confined the plane of the paper.

(a) A test charge $\left(q\right)$, of same sign as $Q$, is placed on the median $AD$ at a point at a distance $\delta$ below $O$. Obtain the force $\left(F\right)$ felt by the test charge.
(b) Assuming $\delta \ll a$ discuss the motion of the test charge when it is released.
(c) Obtain the force $\left(F_D\right)$ on this test charge if it is placed at the point $D$ as shown in the figure.
(d) In the figure below mark the approximate locations of the equilibrium point(s) for this system. Justify your answer.
(e) Is the equilibrium at $O$ stable or unstable if we displace the test charge in the direction of $OP$? The line $PQ$ is parallel to the base $BC$. Justify your answer.
(f) Consider a rectangle $ABCD$. Equal charges are fixed at the vertices $A, B, C$ and $D. O$ is the centroid. In the figure below mark the approximate locations of all the neutral points of the system for a test charge with same sign as the charges on the vertices. Dotted lines are drawn for the reference.
(g) How many neutral points are possible for a system in which $N$ charges are placed at the $N$ vertices of a regular $N$ sided polygon ?.