Electrostatic Potential and Capacitance Test

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Electrostatic Potential and Capacitance Test - 78

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Question 1
1 / -0

One plate of a parallel plate capacitor is suspended from a beam of a physical balance as shown in figure. The area of each plate is $$625 \ cm^2$$ and the distance between these plates is $$5 \ mm$$. If an additional mass $$0.04 \ gm$$ is placed in the other pan of the balance, then the potential difference required between the plates to keep it in equilibrium will be:

A
$$150 \ V$$

B
$$188 \ V$$

C
$$225 \ V$$

D
$$310 \ V$$

Solution

One plate capacitor
parallel capacitor
The area of each plate$$=625cm^2$$
Distance between these plate$$=5mm$$
Additional mass$$=0.04gm$$
$$C=\cfrac{E_0A}{d}\\ \quad=\cfrac{8.854\times(625\times10^{-4})}{5\times10^{-3}}\\ \quad=110.675\\V=Q/C=\cfrac{0.04\times10^{-3}}{110.675}=225V$$
Question 2
1 / -0

A and B are two points in a closed circuit. The potential difference across the condenser of capacity $$5 \mu F$$ is

A
6 V

B
10 V

C
16 V

D
4 V

Solution
Question 3
1 / -0

Five capacitors of $$ 10\mu F$$ capacity each are connected to a d.c. potential of $$100$$ volts as shown in the adjoining figure. The equivalent capacitance between the points $$A$$ and $$B$$ will be equal to :

A
$$ 40 \mu F $$

B
$$ 20 \mu F $$

C
$$ 30 \mu F $$

D
$$ 10 \mu F $$

Solution

$$\textbf{Step 1: Balanced Wheat stone bridge }$$
Given circuit is an example of the wheat stone bridge.
Here the ratio of upper and lower capacitances on both sides is equal i.e. $$\cfrac{10\mu F}{10\mu F} = \cfrac{10\mu F}{10\mu F}$$, therefore this is a balanced wheat stone bridge.
Hence, the potential difference across the middle capacitor will be zero, so we can eliminate the middle capacitor(Refer figure).
$$\textbf{Step 2: Equivalent capacitance }$$
Now solve series and parallel arrangement shown in figure
Firstly solve for series arrangement in upper and lower branches:
$$\dfrac {1}{C}=\dfrac{1}{C_1} +\dfrac{1}{C_2}= \dfrac{1}{10}+\dfrac{1}{10}=\dfrac{1}{5}$$
So, $$C =5 \mu F$$

Now both upper and lower $$C$$ will be in parallel.
So, $$C_{net}=C+C=5+5=10 \mu F$$

Therefore the equivalent capacitance across AB is $$10\,\mu F$$.
Hence, option D is correct.
Question 4
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In the figure given, the effective capacitance between $$A$$ and $$B$$ will be

A
$$C$$

B
$$C/2$$

C
$$2C$$

D
$$3C$$

Solution
Question 5
1 / -0

The equivalent capacitance between the points A and C is given by

A
$$\dfrac{10}{3}$$ C

B
15 C

C
$$\dfrac{3}{10}$$ C

D
20 C

Solution
Question 6
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A parallel-plate capacitor of plate area $$A$$ and plate separation $$d$$ is charged to a potential difference and then the battery is disconnected. A slab of dielectric constant $$K$$ is then inserted between the plates of the capacitor so as to fill the whole space between the plates. Find the work done on the system the process of inserting the slab.

A
$$\dfrac{\varepsilon_0AV^2}{2d}\left({{1}-{\dfrac{1}{K}}}\right)$$

B
$$\dfrac{\varepsilon_0AV^2}{d}\left(\dfrac{1}{K}-1\right)$$

C
$$\dfrac{\varepsilon_0AV^2}{2d}\left(\dfrac{1}{K}+1\right)$$

D
$$\dfrac{\varepsilon_0AV^2}{d}\left(\dfrac{1}{K}+1\right)$$

Solution

after inserting slab

$$new\quad capacitance { C }^{ \prime }=KC=\dfrac { K{ \epsilon_o }A }{ d } $$

$$new\quad potential\quad difference{ V }^{ \prime }=\frac { V }{ K } $$

$$new\quad electricfield\quad E^{ \prime }=\dfrac { { V }^{ \prime } }{ d } =\dfrac { V }{ kd } $$

$$new\quad charge{ Q }^{ \prime }{ C }^{ \prime }=\dfrac { { \epsilon_o }AV }{ d } $$

$$work=final-initial$$

$$=\dfrac { 1 }{ 2 } { C }^{ \prime }{ V }^{ \prime }-\dfrac { 1 }{ 2 } { CV }^{ 2 }$$

$$\dfrac { 1 }{ 2 } \left( KC \right) =\left( 1-\dfrac { 1 }{ K } \right) $$

$$\left[ \omega \right] =\dfrac { { \epsilon_o }{ AV }^{ 2 } }{ 2d } \left( 1-\dfrac { 1 }{ K } \right) $$
Question 7
1 / -0

In the figure show the potential difference between hollow spheres $$A$$ and $$B$$ is:-

A
zero

B
$$- \dfrac{q}{16 \pi \varepsilon_0 R}$$

C
$$\dfrac{q}{16 \pi \varepsilon_0 R}$$

D
$$\dfrac{- q}{24 \pi \varepsilon_0 R}$$

Solution
Question 8
1 / -0

The equivalent capacitance between point $$A$$ and $$B$$ of a combination shown in the figure is

A
$$C$$

B
$$2C$$

C
$$C/2$$

D
$$None\ of\ these$$

Solution
Question 9
1 / -0

The equivalent capacitance between point $$A$$ and $$B$$ if the circuit shown will be:-

A
$$\dfrac{2}{3} \mu F$$

B
$$\dfrac{5}{3} \mu F$$

C
$$\dfrac{8}{3} \mu F$$

D
$$\dfrac{7}{3} \mu F$$

Solution
Question 10
1 / -0

Two parallel plate air capacitors each of capacitance $$C$$ were connected in series to a battery with e.m.f. $$E$$. Then one of the capacitors was filled up with a uniform dielectric with relative permittivity $$k$$. What amount of charge flows through the battery?

A
$$\Delta q=\dfrac{1}{2}CE\dfrac{k+1}{k-1}$$

B
$$\Delta q=\dfrac{1}{2}CE\dfrac{k-1}{k+1}$$

C
$$\Delta q=\dfrac{1}{2}CE\dfrac{2k-1}{k+1}$$

D
None of these