Electrostatic Potential and Capacitance Test

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Electrostatic Potential and Capacitance Test - 35

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Weekly Quiz Competition

Question 1
1 / -0

In the adjoining figure, capacitor $$(1)$$ and $$(2)$$ have a capacitance $$'C'$$ each. When the dielectric of dielectric constant $$K$$ is inserted between the plates of one of the capacitor, the total charge flowing through battery is :

A
$$\displaystyle\frac{KCE}{K+1}\space from\space B\space to\space C$$

B
$$\displaystyle\frac{KCE}{K+1}\space from\space C\space to\space B$$

C
$$\displaystyle\frac{(K-1)CE}{2(K+1)}\space from\space B\space to\space C$$

D
$$\displaystyle\frac{(K-1)CE}{2(K+1)}\space from\space C\space to\space B$$

Solution

Before insert dielectric, the equivalent capacitance $$C_{eq}= \frac{C\times C}{C+C}=C/2$$
$$Q_{eq}=C_{eq}E=CE/2$$
As battery is connected to capacitor so E remains constant.
After insert dielectric , $$C'_{eq}=\dfrac{KC C}{KC+C}=\dfrac{K}{K+1}C$$
$$Q'_{eq}=C'_{eq}E=\dfrac{K}{K+1}CE$$
Charge flow through the battery is $$\Delta Q=Q'_{eq}-Q_{eq}=\frac{K}{K+1}CE-\dfrac{CE}{2}=\dfrac{(K-1)CE}{2(K+1)}$$
As E is constant so for inserting dielectric the charge on the capacitor will increase and it will flow from capacitance to battery to maintain constant $$E$$
Question 2
1 / -0

Four identical plates $$1, 2, 3,$$ and $$4$$ are placed parallel to each other at equal distance as shown in the figure. Plates $$1$$ and $$4$$ are joined together and the space between $$2$$ and $$3$$ is filled with a dielectric of dielectric constant $$k = 2$$. The capacitance of the system between $$1$$ and $$3$$ & $$2$$ and $$4$$ are $$C_1$$ and $$C_2$$ respectively. The ratio $$C_1/C_2$$ is :

A
$$\displaystyle\frac{5}{3}$$

B
$$1$$

C
$$\displaystyle\frac{3}{5}$$

D
$$\displaystyle\frac{5}{7}$$

Solution

The capacitance of parallel plate capacitor is $$C=\dfrac{A\epsilon_0}{d}$$
The capacitance of parallel plate capacitor with dielectric is $$C'=\dfrac{Ak\epsilon_0}{d}=kC=2C$$
Here 1 and 2 plate and 3 and 4 plate will make the capacitor of same capacitance $$C$$ and 2 and 3 will make the capacitance of $$C'$$.
Here the capacitance in between 1 and 3 is $$C_1=\dfrac{CC'}{C+C'}=\dfrac{C.2C}{C+2C}=2C/3$$
and the capacitance in between 2 and 4 is $$C_2=\dfrac{CC'}{C+C'}=\dfrac{C.2C}{C+2C}=2C/3$$
$$\therefore \frac{C_1}{C_2}=1$$
Question 3
1 / -0

A charge $$3$$ coulomb experiences a force $$3000\ N$$ when placed in a uniform electric field. The potential difference between two points seperated by a distance of $$1\ cm$$ along the field lines is :

A
$$10\ V$$

B
$$90\ V$$

C
$$1000\ V$$

D
$$9000\ V$$

Solution

Electric force $$F=qE$$

given, $$F=3000 N, q=3 C$$

$$\therefore E=\dfrac{F}{q}=\dfrac{3000}{3}=1000 V/m$$

Potential difference, $$V={E}{d}$$ where $$d=0.01 m$$

$$\therefore V=1000\times 0.01=10 V$$

Ans:(A)
Question 4
1 / -0

The minimum number of capacitors each of $$3\space\mu F$$ required to make a circuit with an equivalent capacitance $$2.25\space\mu F$$ is :

A
$$3$$

B
$$4$$

C
$$5$$

D
$$6$$

Solution

There are four capacitors in which 3 capacitors are parallel and in series with 1 capacitor.
Net Capacitance for 3 parallel capacitors = 9 $$\mu F$$
Equivalent Capacitance for 3 parallel capacitors and in series with 1 capacitor = $$\dfrac{9 \times 3}{9+3} = 2.25$$ $$\mu F$$
Question 5
1 / -0

Statement 1: The electrostatic force between the plates of a charged isolated capacitor decreases when dielectric fills whole space between plates.

Statement 2: The electric field between the plates of a charged isolated capacitance decreases when dielectric fills whole space between plates

A
Statement-1 is true, Statement-2 is true and Statement-2 is the correct explanation for Statement-1

B
Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1

C
Statement-1 is true, Statement-2 is false

D
Statement-1 is false, Statement-2 is true

Solution

For isolated capacitor the charge (Q) remains constant.
The field between the plates without dielectric is $$E=\frac{Q}{A\epsilon_0}$$
The field between the plates with dielectric is $$E'=\frac{Q}{Ak\epsilon_0}=E/k$$ where k is dielectric constant. Thus the filed between the plates will decrease.
The force between the plates do not depend on the field between the plates and its depend on the filed due to a charge plate.
Question 6
1 / -0

A electric charge $${ 10 }^{ -8 }C$$ is placed at the point $$(4m,7m,2m)$$. At the point $$(1m,3m,2m)$$ the electric

A
potential will be $$18V$$

B
field has no $$Y$$-component

C
field will be along $$Z$$-axis

D
potential will be $$1.8V$$

Solution

Electric potential$$,\ V=\dfrac{1}{4\pi \epsilon_0}\dfrac{q}{r}=\dfrac{1}{4\pi \epsilon_0}\dfrac{q}{\sqrt{(4-1)^2+(7-3)^2+(2-2)^2}}=\dfrac{10^{-8}}{4\pi \times 8.854\times 10^{-12}\times 5}=18 volt$$
Question 7
1 / -0

Two capacitors $$C_1$$ and $$C_2$$ are connected in series, assume that $$C_1 < C_2$$. The equivalent capacitance of this arrangement is $$C$$, where

A
$$C < C_1/2$$

B
$$C_1/2 < C < C_1$$

C
$$C_1 < C < C_2$$

D
$$C_2 < C < 2C_2$$

Solution

We have series capacitance,

$$ \frac {1}{C} = \frac {1}{C_1} + \frac {1}{C_2}$$

$$\implies C = \frac {C_1 C_2}{C_1 + C_2}$$

And also given that $$C_2 > C_1$$

Hence, $$2C_2 > C_2 > C_1 > C > \frac{C_1}{2}$$

Option $$(a)$$, $$(c)$$ and $$(d)$$ are wrong.
Question 8
1 / -0

A dense sphere of mass $$M$$ is placed at the centre of a circle of radius $$R$$. Find the work done, when a particle of mass $$m$$ is brought from A to B along a circle as shown in the figure.

A
$$zero$$

B
$$\displaystyle \frac{G M m}{R}$$

C
$$\displaystyle - \frac{G Mm}{R}$$

D
$$\displaystyle \frac{2 G M m}{R}$$

Solution

$$E = PE + KE$$
From energy conservation,
$$\dfrac{1}{2}mv_A ^2 + (PE)_A= \dfrac{1}{2}mv_B ^2 + (PE)_B$$
Since speed does not change from A to B, KE remains same.
Distance of the particle at A and B is equal to the radius of the the circle.
$$PE = -\dfrac{GMm}{R}$$
Hence, $$(PE)_A = (PE)_B$$
Hence, work done is zero as there is no change in total energy E from A to B.
Question 9
1 / -0

An electrical charge of $$2$$ $$\mu$$C is placed at the point $$(1, 2, 3)$$. At the point $$(2, 3, 4)$$ the electric field and potential will be :

A
$$6 \times 10^3 NC^{-1}$$ and $$6 \times 10^3 JC^{-1}$$

B
$$6000 NC^{-1}$$ and $$6000 \sqrt{3} JC^{-1}$$

C
$$6 \times 10^3 NC^{-1}$$ and $$3\sqrt{3} JC^{-1}$$

D
none of the above

Solution

The electric field , $$\displaystyle E=\dfrac{1}{4\pi\epsilon_0} \dfrac{q}{r^2}=9\times 10^9\times \dfrac{2\times 10^{-6}}{(2-1)^2+(3-2)^2+(4-3)^2}=6000 NC^{-1}$$

and potential, $$\displaystyle V=\dfrac{1}{4\pi\epsilon_0} \dfrac{q}{r}=9\times 10^9\times \dfrac{2\times 10^{-6}}{\sqrt{(2-1)^2+(3-2)^2+(4-3)^2}}=6000 \sqrt 3 NC^{-1}$$
Question 10
1 / -0

Which of the following combinations of seven identical capacitors each of 2$$\mu$$F gives a capacitance of 10/11 $$\mu$$F?

A
5 in parallel with 2 in series

B
4 in parallel with 3 in series

C
3 in parallel with 4 in series

D
2 in parallel with 5 in series

Solution

A) $$\displaystyle \frac{1}{C_{eq}}=\frac{1}{5C}+\frac{1}{C}+\frac{1}{C}=\frac{11}{5C}=\frac{11}{5\times 2}=11/10\Rightarrow C_{eq}=10/11 \mu F$$
B) $$\displaystyle \frac{1}{C_{eq}}=\frac{1}{4C}+\frac{1}{C}+\frac{1}{C}+\frac{1}{C}=\frac{13}{4C}=\frac{13}{4\times 2}=13/8\Rightarrow C_{eq}=8/13 \mu F$$
C) $$\displaystyle \frac{1}{C_{eq}}=\frac{1}{3C}+\frac{1}{C}+\frac{1}{C}+\frac{1}{C}+\frac{1}{C}=\frac{13}{3C}=\frac{13}{3\times 2}=13/6\Rightarrow C_{eq}=6/13 \mu F$$
D) $$\displaystyle \frac{1}{C_{eq}}=\frac{1}{2C}+\frac{1}{C}+\frac{1}{C}+\frac{1}{C}+\frac{1}{C}+\frac{1}{C}=\frac{11}{2C}=\frac{11}{2\times 2}=11/4\Rightarrow C_{eq}=4/11 \mu F$$

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Electrostatic Potential and Capacitance Test - 35

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Electrostatic Potential and Capacitance Test - 35

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Question 1

$$\displaystyle\frac{KCE}{K+1}\space from\space B\space to\space C$$

$$\displaystyle\frac{KCE}{K+1}\space from\space C\space to\space B$$

$$\displaystyle\frac{(K-1)CE}{2(K+1)}\space from\space B\space to\space C$$

$$\displaystyle\frac{(K-1)CE}{2(K+1)}\space from\space C\space to\space B$$

Solution

Question 2

$$\displaystyle\frac{5}{3}$$

$$1$$

$$\displaystyle\frac{3}{5}$$

$$\displaystyle\frac{5}{7}$$

Solution

Question 3

$$10\ V$$

$$90\ V$$

$$1000\ V$$

$$9000\ V$$

Solution

Question 4

$$3$$

$$4$$

$$5$$

$$6$$

Solution

Question 5

Statement-1 is true, Statement-2 is true and Statement-2 is the correct explanation for Statement-1

Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1

Statement-1 is true, Statement-2 is false

Statement-1 is false, Statement-2 is true

Solution

Question 6

potential will be $$18V$$

field has no $$Y$$-component

field will be along $$Z$$-axis

potential will be $$1.8V$$

Solution

Question 7

$$C < C_1/2$$

$$C_1/2 < C < C_1$$

$$C_1 < C < C_2$$

$$C_2 < C < 2C_2$$

Solution

Question 8

$$zero$$

$$\displaystyle \frac{G M m}{R}$$

$$\displaystyle - \frac{G Mm}{R}$$

$$\displaystyle \frac{2 G M m}{R}$$

Solution

Question 9

$$6 \times 10^3 NC^{-1}$$ and $$6 \times 10^3 JC^{-1}$$

$$6000 NC^{-1}$$ and $$6000 \sqrt{3} JC^{-1}$$

$$6 \times 10^3 NC^{-1}$$ and $$3\sqrt{3} JC^{-1}$$

none of the above

Solution

Question 10

5 in parallel with 2 in series

4 in parallel with 3 in series

3 in parallel with 4 in series

2 in parallel with 5 in series

Solution

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