Increasing Capacitance Even More

Q. A solid non conducting sphere of radius $R$ is charged uniformly. At what distance from its surface is the electrostatic potential becomes half of the potential at the centre?

1. $\frac{4R}{3}$
2. $\frac{R}{2}$
3. $\frac{R}{3}$
4. $2\text{R}$

Q. A thin spherical conducting shell of radius $R$ has a charge $q$. Another charge $Q$ is placed at the center of the shell. The electrostatic potential at a point $P$ at a distance $R/2$ from the center of the shell is

1. $\frac{2Q}{4\pi {\epsilon }_{0}R}$
2. $\frac{2Q}{4\pi {\epsilon }_{0}R}-\frac{q}{4\pi {\epsilon }_{0}R}$
3. $\frac{2Q}{4\pi {\epsilon }_{0}R}+\frac{q}{4\pi {\epsilon }_{0}R}$
4. $\frac{2Q}{4\pi {\epsilon }_{0}R}+\frac{2q}{4\pi {\epsilon }_{0}R}$

Q.

Which orientation of an electric dipole in a uniform electric field would correspond to stable equilibrium?

Q. A 3uF capacitor is charged to a potential of 300v and 2uF capacitor is charged to 200.The capacitors are then connected in parallel with plates of opposite polarities joined together.what amount of charge will flow, when the plates are so connected?

Q. If atmospheric electric field is approximately $150\text{V/m}$ and radius of the earth is $6400\text{km}$, then total charge on the earth's surface is

1. $6.8\times {10}^5\text{C}$
2. $6.8\times {10}^6\text{C}$
3. $6.8\times {10}^4\text{C}$
4. $6.8\times {10}^9\text{C}$

Q. Charges $+q$ and $-q$ are placed at points $\text{A}$ and $\text{B}$ respectively which are at a distance $2L$ apart and $\text{C}$ is the mid point between $\text{A}$ and $\text{B}$. The workdone in moving a charge $+Q$ along the semicircle $\text{CRD}$ with $B$ as centre is

1. $\frac{qQ}{4\pi {\epsilon }_{0}L}$
2. $\frac{qQ}{6\pi {\epsilon }_{0}L}$
3. $\frac{qQ}{2\pi {\epsilon }_{0}L}$
4. $-\frac{qQ}{6\pi {\epsilon }_{0}L}$

Q. The nuclear charge $\left(Ze\right)$ is non-uniformly distributed within a nucleus of radius $R$. The charge density $\rho \left(r\right)$( charge per unit volume) is dependent only on the radial distance $r$ from the centre of the nucleus as shown in figure. The electric field is only along the radial direction. The electric field within the nucleus is generally observed to be linearly dependent on $r$. This implies

1. $a=0$
2. $a=\frac{R}{2}$
3. $a=R$
4. $a=\frac{2R}{3}$

Q. For the given uniformly charged ring, magnitude of the net electric field at point $\text{P}$ is

1. $11.3\text{N/C}$
2. $21.3\text{N/C}$
3. $31.3\text{N/C}$
4. $41.3\text{N/C}$

Q. Three different dielectrics are filled in a parallel plate capacitor as shown. What should be the value of dielectric constant of a material , which when fully filled between the plates produces same capacitance ? (Asssume terminals are connected between $AC$ and $BD$).

1. $4$
2. $6$
3. $5$
4. $9$

Q. A sphere of radius R carries charge such that its volume charge density is proportional to the square of the distance from the centre. What is the ratio of the magnitude of the electric field at a distance 2R from the centre to the magnitude of the electric field at a distance of $\frac{R}{2}$ from the centre $(i.e.,E_{r=2R}/E_{r=R/2})$?

1. 1
2. 2
3. 4
4. 8

Q. Find the equivalent capacitance between $\text{A}$ and $\text{B}$ of the circuit shown in the figure.


Given:
$C_1=5\mu \text{F};C_2=20\mu \text{F};C_3=10\mu \text{F};C_4=40\mu \text{F};C_5=30\mu \text{F}$

1. $12\mu \text{F}$
2. $24\mu \text{F}$
3. $6\mu \text{F}$
4. $21\mu \text{F}$

Q. The total charge enclosed in an incremental volume of $2\times {10}^{-9}{\text{m}}^3$ located at the origin is$\text{nC}$, if electric flux density of its field is found as $D=e^{-x}\sin \hat{i}-e^{-x}\cos y\hat{j}+2z\hat{k}{\text{C/m}}^2$.

Q. A conducting sphere of radius $R$ is given a charge $Q$. The electric potential and the electric field at the center of the sphere respectively are

1. Zero and $\frac{Q}{4\pi {ϵ}_0{R}^2}$
2. $\frac{Q}{4\pi {ϵ}_{0}R}$ and Zero
3. $\frac{Q}{4\pi {ϵ}_{0}R}$ and $\frac{Q}{4\pi {ϵ}_0{R}^2}$
4. Both are Zero

Q. Consider a conducting spherical shell of radius $R$ with its center at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field $|\overrightarrow{E}\left(r\right)|$ and the electric potential $V\left(r\right)$ with the distance $r$ from the center, is best represented by the graph
(Here dotted line represents potential curve and bold line represents electric field curve):

1. 
2. 
3. 
4. 

Q. The ratio of electric field due to an electric dipole on its axis and on the perpendicular bisector of dipole is

Q. Two charged conducting spheres of radii $r_1$ and $r_2$ have same electric field near their surfaces. The ratio of their electric potential is

1. $r_1^2/r_2^2$
2. $r_2^2/r_1^2$
3. $r_1/r_2$
4. $r_2/r_1$

Q. Two conducting spheres of radii $R_1$ and $R_2$ are charged with charges $Q_1$ and $Q_2$. If they are brought in contact, there is

1. no increase in the energy of the system
2. an increase in the energy of the system if $Q_1{R}_2\ne Q_2{R}_1$
3. always a decrease in the energy of system
4. an decrease in the energy of system if $Q_1{R}_2\ne Q_2{R}_1$

Q. The electric potential at the centre of hemisphere of radius $R$ having uniform surface charge density $\sigma$ is

1. $\frac{\sigma R}{{ϵ}_0}$
2. $\frac{2\sigma R}{{ϵ}_0}$
3. $\frac{\sigma R}{2{ϵ}_0}$
4. $\frac{\sigma R}{3{ϵ}_0}$

Q. An annular disc has inner and outer radius $R_1$​ and $R_2$​ respectively. Charge is uniformly distributed. Surface charge density is $\sigma$. Find the electric field at any point distant $y$ along the axis of the disc.

1. $\frac{\sigma y}{2{ϵ}_0}[\frac{1}{\sqrt{R_1^2+y^2}}-\frac{1}{\sqrt{R_2^2+y^2}}]$
2. $\frac{\sigma }{2{ϵ}_0}$
3. $\frac{\sigma y}{2{ϵ}_0(R_2-R_1)}$
4. $\frac{\sigma }{2{ϵ}_0}\log \frac{R_2+y}{R_1+y}$

Q. A capacitor has a capacity $C$ and reactance $X$. If the capacitance and frequency both are doubled, the reactance will become

1. 
   $4X$
2. 
   $X/2$
3. 
   $\frac{X}{4}$
4. 
   $2X$

Q. Inside a conducting spherical shell of inner radius $3R$ and outer radius $5R$ , a point charge $Q$ is placed at a distance $R$ from the centre of the shell. The electric potential at the centre of the shell will be.

1. $\frac{1}{4\pi {ϵ}_0}\frac{Q}{R}$
   
2. $\frac{1}{4\pi {ϵ}_0}\frac{5Q}{6R}$
3. $\frac{1}{4\pi {ϵ}_0}\frac{13Q}{15R}$
4. $\frac{1}{4\pi {ϵ}_0}\frac{7Q}{9R}$

Q.

Charge on the outer sphere is $q$ and the inner sphere is grounded. Then the charge $q^{\prime }$ on the inner sphere is, for $(r_2>r_1)$

1. zero
2. $q^{\prime }=q$
3. $q^{\prime }=-\frac{r_1}{r_2}q$
4. $q^{\prime }=\frac{r_1}{r_2}q$

Q. $\text{A}$ and $\text{B}$ are two concentric spherical shells. If $\text{A}$ is given a charge $+q$ while $\text{B}$ is earthed as shown in figure, then $[k=\frac{1}{4\pi {\epsilon }_0}]$

1. charge on the outer surface of shell $\text{B}$ is zero.
2. the charge on $\text{B}$ is equal and opposite to that of $\text{A}$.
3. the field inside $\text{A}$ and outside $B$ is zero .
4. the field outside $\text{A}$ has a finite non-zero value.

Q. An annular disc of inner radius $a$ and outer radius $2a$ is uniformly charged with uniform surface charge density $\sigma$. Find the potential at a distance $a$ from the centre at a point $P$ lying on the axis which is perpendicular to the plane containing the disc.

1. $\frac{\sigma }{{ϵ}_0}a(\sqrt{5}-\sqrt{2})$
   
2. $\frac{\sigma }{2{ϵ}_0}a(\sqrt{5}-\sqrt{2})$
3. $\frac{\sigma }{2{ϵ}_0}(\sqrt{3}-\sqrt{2})$
   
4. $$\frac{\sigma }{2{ϵ}_0}a$$
   

Q. At a distance of $5cm$ and $10cm$ outward from the surface of a uniformly charged solid sphere, the potentials are $100V$ and $75V$, respectively. Then

1. potential at its surface is $150V$
2. the charge on the sphere is $\left(\frac{5}{3}\right)\times {10}^{10}C$
3. the electric field on the surface is $1500V{m}^{-1}$
   
4. the electric potential at its center is $0V$

Q. Half ring of radius $a$ is uniformly charged with linear charge density $\lambda$. Find out electric field intensity at point $O$.

1. $\frac{\lambda }{2\pi {ϵ}_{0}a}(-\hat{i})$
2. $\frac{\lambda }{4\pi {ϵ}_{0}a}(+\hat{i})$
3. $\frac{\lambda }{2\pi {ϵ}_{0}a}(+\hat{i})$
4. $\frac{\lambda }{4\pi {\epsilon }_{0}a}(-\hat{i})$

Q. Capacitance of a capacitor when a dielectric is introduced between its plates.

1. increases
2. decreases
3. becomes zero

Q. For the given uniformly charged quarter ring, net electric field at the center is

1. $140\text{N/C}$
2. $160\text{N/C}$
3. $180\text{N/C}$
4. $200\text{N/C}$

Q. A solid conducting sphere having a charge $Q$ is surrounded by an uncharged concentric conducting spherical shell. Let the potential difference between the surface of the solid sphere and the outer surface of the shell be $V.$ If the shell is now given a charge $-3Q,$ the new potential difference between the same two surfaces is:

1. $V$
2. $2V$
3. $4V$
4. $-2V$

Q. Assertion. If a metal sheet is introduced in between plates of a parallel plate capacitor, then capacitance increases Reason. On insertion of the metal between the plates of a parallel plate capacitor electric field always decreases