In figure two positive charges $q_2$ and $q_3$ fixed along the $y$-axis, exert a net electric force in the $+x$-direction on a charge $q_1$ fixed along the $x$-axis. If a positive charge $Q$ is added at $(x, 0)$, the force on $q_1$

A point positive charge is brought near an isolated conducting sphere (figure). The electric field is best given by

The electric flux through the surface

Five charges $q_1, q_2, q_3, q_4$, and $q_5$ are fixed at their positions as shown in Figure, $S$ is a Gaussian surface. The Gauss' law is given by $\int_S E . d S=\frac{q}{\varepsilon_0}$. Which of the following statements is correct?

Figure shows electric field lines in which an electric dipole $P$ is placed as shown. Which of the following statements is correct?

A point charge $+q$ is placed at a distance $d$ from an isolated conducting plane. The field at a point $P$ on the other side of the plane is

A hemisphere is uniformely charged positively. The electric field at a point on a diameter away from the centre is directed

If $\int_S \mathbf{E} . \mathrm{dS}=0$ over a surface, then

The electric field at a point is

If there were only one type of charge in the universe, then

Consider a region inside which there are various types of charges but the total charge is zero. At points outside the region,

Refer to the arrangement of charges in figure and a Gaussian surface of radius $R$ with $Q$ at the centre. Then,

A positive charge $Q$ is uniformly distributed along a circular ring of radius R.A small test charge $q$ is placed at the centre of the ring figure. Then,

An arbitrary surface encloses a dipole. What is the electric flux through this surface?

A metallic spherical shell has an inner radius $R_1$ and outer radius $R_2$. A charge $Q$ is placed at the centre of the spherical cavity. What will be surface charge density on

(i) the inner surface

(ii) the outer surface?

The dimensions of an atom are of the order of an Angstrom. Thus, there must be large electric fields between the protons and electrons. Why, then is the electrostatic field inside a conductor zero?

If the total charge enclosed by a surface is zero, does it imply that the electric field everywhere on the surface is zero? Conversely, if the electric field everywhere on a surface is zero, does it imply that net charge inside is zero.

Sketch the electric field lines for a uniformly charged hollow cylinder shown in figure.

What will be the total flux through the faces of the cube as given in the figure with side of length $a$ if a charge $q$ is placed at?

(a) $A$ a corner of the cube

(b) $B$ mid-point of an edge of the cube

(c) $C$ centre of a face of the cube

(d) $D$ mid-point of $B$ and $C$

A paisa coin is made up of $\mathrm{Al}-\mathrm{Mg}$ alloy and weight 0.75 g . It has a square shape and its diagonal measures 17 mm . It is electrically neutral and contains equal amounts of positive and negative charges.

Consider a coin of Question 20. It is electrically neutral and contains equal amounts of positive and negative charge of magnitude 34.8 kC . Suppose that these equal charges were concentrated in two point charges separated by

(i) $1 \mathrm{~cm}\left(\sim \frac{1}{2} \times\right.$ diagonal of the one paisa coin $)$

(ii) 100 m ( length of a long building)

(iii) $10^6 \mathrm{~m}$ (radius of the earth). Find the force on each such point charge in each of the three cases. What do you conclude from these results?

Figure represents a crystal unit of cesium chloride, CsCl. The cesium atoms, represented by open circles are situated at the corners of a cube of side 0.40 nm , whereas a Cl atom is situated at the centre of the cube. The Cs atoms are deficient in one electron while the Cl atom carries an excess electron.

(i) What is the net electric field on the Cl atom due to eight Cs atoms?

(ii) Suppose that the Cs atom at the corner $A$ is missing. What is the net force now on the Cl atom due to seven remaining Cs atoms?

Two charges $q$ and $-3 q$ are placed fixed on $x$-axis separated by distance d. Where should a third charge $2 q$ be placed such that it will not experience any force?

Figure shows the electric field lines around three point charges A, B and C

(i) Which charges are positive?

(ii) Which charge has the largest magnitude? Why?

(iii) In which region or regions of the picture could the electric field be zero? Justify your answer.

(a) Near $A$

(b) Near $B$

(c) Near C

(d) Nowhere

Five charges, q each are placed at the corners of a regular pentagon of side.

(a) (i) What will be the electric field at 0 , the centre of the pentagon?

(ii) What will be the electric field at $O$ if the charge from one of the corners (say $A$ ) is removed?

(iii) What will be the electric field at $O$ if the charge $q$ at $A$ is replaced by $-q$ ?

(b) How would your answer to (a) be affected if pentagon is replaced by $n$-sided regular polygon with charge $q$ at each of its corners?

In 1959 Lyttleton and Bondi suggested that the expansion of the universe could be explained if matter carried a net charge. Suppose that the universe is made up of hydrogen atoms with a number density $N$, which is maintained a constant. Let the charge on the proton be $e_p=-(1+y) e$ where $e$ is the electronic charge.

(a) Find the critical value of $y$ such that expansion may start.

(b) Show that the velocity of expansion is proportional to the distance from the centre.

Consider a sphere of radius $R$ with charge density distributed as $p(r)=k r$ for $r \leq R=0$ for $r>R$.

(a) Find the electric field as all points $r$.

(b) Suppose the total charge on the sphere is $2 e$ where $e$ is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution.

Two fixed, identical conducting plates ( $\alpha$ and $\beta$ ), each of surface area $S$ are charged to $-Q$ and $q$, respectively, where $Q>q>0$. A third identical plate $(\gamma)$, free to move is located on the other side of the plate with charge $q$ at a distance $d$ (figure). The third plate is released and collides with the plate $\beta$. Assume the collision is elastic and the time of collision is sufficient to redistribute charge amongst $\beta$ and $\gamma$.

(a) Find the electric field acting on the plate $\gamma$ before collision.

(b) Find the charges on $\beta$ and $\gamma$ after the collision.

(c) Find the velocity of the plate $\gamma$ after the collision and at a distance $d$ from the plate $\beta$.

There is another useful system of units, besides the SI/MKS. A system, called the CGS (Centimeter-Gram-Second) system. In this system, Coulomb's law is given by $\mathbf{F}=\frac{Q q}{r^2} \hat{\mathbf{r}}$. where the distance $r$ is measured in $\mathrm{cm}\left(=10^{-2} \propto\right), F$ in dynes $\left(=10^{-5} \mathrm{~N}\right)$ and the charges in electrostatic units (es units), where 1 es unit of charge $=\frac{1}{[3]} \times 10^{-9} \mathrm{C}$. The number [3] actually arises from the speed of light in vacuum which is now taken to be exactly given by $c=2.99792458 \times 10^8 \mathrm{~m} / \mathrm{s}$. An approximate value of $c$, then is $c=3 \times 10^8 \mathrm{~m} / \mathrm{s}$.

(i) Show that the Coulomb's law in CGS units yields 1 esu of charge $=1(\text { dyne })^{1 / 2} \mathrm{~cm}$. Obtain the dimensions of units of charge in terms of mass $M$, length $L$ and time $T$. Show that it is given in terms of fractional powers of $M$ and $L$.

(ii) Write 1 esu of charge $=x C$, where $x$ is a dimensionless number. Show that this gives $\frac{1}{4 \pi \varepsilon_0}=\frac{10^{-9}}{\mathrm{x}^2} \frac{\mathrm{Nm}^2}{\mathrm{C}^2}$. With $x=\frac{1}{[3]} \times 10^{-9}$, we have $\frac{1}{4 \pi \varepsilon_0}=[3]^2 \times 10^9 \frac{\mathrm{Nm}^2}{\mathrm{C}^2}, \frac{1}{4 \pi \varepsilon_0}=(2.99792458)^2 \times 10^9 \frac{\mathrm{Nm}^2}{\mathrm{C}^2}$ (exactly).

Two charges $-q$ each are fixed separated by distance $2 d$. A third charge $q$ of mass $m$ placed at the mid-point is displaced slightly by $x(x< d)$ perpendicular to the line joining the two fixed charged as shown in figure. Show that $q$ will perform simple harmonic oscillation of time period.

$T=\left[\frac{8 \pi^3 \varepsilon_0 m d^3}{q^2}\right]^{1 / 2}$

Total charge $-Q$ is uniformly spread along length of a ring of radius $R$.A small test charge $+q$ of mass $m$ is kept at the centre of the ring and is given a gentle push along the axis of the ring.

(a) Show that the particle executes a simple harmonic oscillation.

(b) Obtain its time period.