# Multiple Choice Questions on Capacitors and capacitance

Question 1
Capacitance of a parallel plate air capacitor depends on
1. Thickness of conducting plates
2. Charge on the conducting plates
3. Area of the conducting plates
4. Distance of separation between the conducting plates

For parallel plate air capacitor
C = (ε0A)/d
Which clearly shows that capacitance does not depend on charge on both the conducting plates. Hence answer is
(a), (c) and (d)

Question 2
Equivalent capacitance of system of capacitors shown below in the figure is

1. C/2
2. 2C
3. C
4. None of these

Equivalent capacitance of two capacitors each having capacitance C are connected in series combination
$\frac {1}{C^'} =\frac {1}{C} + \frac {1}{C} =\frac {2}{C}$
Or,
$C^' = \frac {C}{2}$
Now circuit given in question takes the form

All the three capacitors in above figure are connected in parallel combination
Thus,
C”=C/2 +C/2 + C = 2C
This is the required equivalent capacitance.
Hence (b) is the correct choice

Question 3
A parallel plate air capacitor with no dielectric between the plates is connected to the constant voltage source. How would capacitance and charge change if dielectric of dielectric constant K=2 is inserted between the plates. C0 and Q0 are the capacitance and charge of the capacitor before the introduction of the dielectric.
1. C=C0/2 ; Q=2Q0
2. C=2C0 ; Q=Q0/2
3. C=C0/2 ; Q=Q0/2
4. C=2C0 ; Q=2Q0

Given that C0 and Q0 are the capacitance and charge of the capacitor before the introduction of the dielectric. Now in the presence of the dielectric capacitance of capacitor becomes
C = (εA)/d
But we know that relative permittivity of any material is given by
ε=Kε0
Where K is dielectric constant of the material.
Thus,
C = (Kε0A)/d
= K C0
Or, C= 2 C0
Again charge on the capacitor after the introduction of dielectric is given as
Q=CV                                                 (a)
And before the introduction of dielectric
Q0= C0V
Now putting the value of C in equation (a) we find
Q=2 C0V
Or, Q=2 Q0
Where V is same before and after the introduction of dielectric as voltage source used is a constant voltage source.
Hence (d) is the correct choice

Question 4
A parallel plate capacitor of capacitance C is connected to a battery and is charged to a potential difference V. Another capacitor of capacitance 2C is similarly charged to a potential difference 2V. The charging battery is then disconnected and the capacitors are connected in parallel to each other in such a way that the positive terminal of one is connected to the negative terminal of the other. The final energy of the configuration is
1. Zero
2. 3CV2/2
3. 25CV2/6
4. 9CV2/2

Charge accumulated on the first capacitor Q1=CV and charge accumulated on second capacitor is Q= (2C) x (2V) = 4CV. Since the capacitors are connected in parallel such that the plates of opposite polarity are connected together, the common potential of the whole system is
$V^' = \frac {Q_2 -Q_1}{C_1 + C_2} = \frac {4CV -CV}{C+2C} =V$
Equivalent capacitance C’=C+2C=3C. hence the final energy of the configuration is
U’=C’V’2/2 = 3CV2/22C)x(2V capacitor is Qe first capacitor
Hence (b) is the correct choice

Question 5
A parallel plate air capacitor is connected to a battery. The quantities charge, voltage, electric field, and energy associated with this capacitor are given by Q0, V0, E0 and U0 respectively. A dielectric slab is now introduced to fill the space between the plates with battery still in connection. The corresponding quantities are now given by Q, V, E and U are related to previous quantities as
1. Q> Q0
2. V> V0
3. E> E0
4. U> U0

The potential difference between the plates remains unchanged after the introduction of dielectric because capacitor remains connected to the battery. Thus V remains equal to V0. Introduction of dielectric increases the capacitance C and hence charge stored in the capacitor also increases since Q=CV thus Q>Q0. Since plate distance and V remains unchanged this implies Electric field E=V/d also remains unchanged. Energy stored in the capacitor increases as it depends on the capacitance C of the capacitor since U=CV2/2. Thus U>U0.
Hence (a) and (d) is the correct choice<

Question 6
The effective capacitance of a capacitor is reduced when capacitors are connected in
1. series
2. parallel
3. series-parallel combination
4. none of the above

Equivalent capacitance of two capacitors each having capacitance C are connected in series combination  $\frac {1}{C^'} =\frac {1}{C_1} + \frac {1}{C_2}$

from this it is clear that the effective capacitance of a capacitor is reduced when capacitors are connected in series combination.
Hence (a) is the correct choice

Question 7
Two identical conducting plates of plate area A are separated by a distance d to form a parallel plate air capacitor. Now a metal sheet of thickness d/2 is inserted between the plates of the capacitor. The ratio of capacitance before the insertion of plate and after the insertion of plate is
1. 1:2
2. 1:1
3. 1:2
4. √2:1

Now capacitance of capacitor before inserting metal plate is
C = (ε0A)/d
And after the insertion of any dielectric plate filling the capacitor partially is
C = (ε0A)/[(d-t)+t/K]
We know that K for a metal is infinite thus
C = (ε0A)/(d-t)
Here t=d/2 thus
C’=(2ε0A)/d=2C
C:C’=1:2
Hence (a) is the correct choice

Question 8
The plates of a parallel plate air capacitor are charged to 100 V. A 2mm plate is inserted between the plates of the capacitor. To maintain the same potential difference , the distance between the capacitor plates is increased by 1.6mm. The dielectric constant of the plate is
1. 1.25
2. 5
3. 2.5
4. 4

When the dielectric slab of thickness t and dielectric constant K is inserted between the plates of the capacitor, the potential difference between the plates of the capacitor is given by
V=E[d-t-(t/K)]
In order to maintain the same value of V separation between the plates should be increased by d’ given by
d’=t(1-1/K) or K=t/(t-d’) = 2mm/(2mm-1.6mm) = 5mm
Hence (b) is the correct choice

Question 9
Four capacitors are connected as shown below in the figure

Here C1=2μF , C2=3μF , C3=4μF and C4=5μF. The equivalent capacitance between A and B is
1. 1.245 μF
2. 4.446 μF
3. 9 μF
4. 3.27 μF

As clear from the figure capacitors C1 and series combination of  C2 , C3 and C4 are connected in parallel. If C’ is the capacitance of series combination then,
$\frac {1}{C^"} = \frac {1}{C_2} + \frac {1}{C_3} + \frac {1}{C_4} = \frac {1}{3} + \frac {1}{4} + \frac {1}{5} = \frac {47}{60}$
or
$C_" =\frac {60}{47} \mu F$
Hence equivalent capacitance between A and B is
C=C1+C’=(2+60/47)μF=3.27 μF
Hence (d) is the correct choice

Question 10
Consider the figure and value of capacitances given in the Question 9 if a 4Volt battery is connected between points A and B total charge stored on the capacitors would be
1. 13.8μC
2.  10 μC
3. 22 μC
4. 4 μC

We have found earlier that equivalent capacitance between points A and B is C=3.27 μF. Total charge stored would be
Q=CV=3.27 μF x 4Volt = 13.8μC
Hence (a) is the correct choice

Question 11
In an isolated parallel plate capacitor of capacitance C, the four surface have charges $Q_1$, $Q_2$, $Q_3$ and $Q_4$ as shown. The potential difference between the plates is

(a) $\frac {Q_1 + Q_2 + Q_3 + Q_4}{2C}$
(b) $\frac {Q_2 + Q_3}{2C}$
(c) $\frac {Q_2 - Q_3}{2C}$
(d) $\frac {Q_1 + Q_4}{2C}$

Plane conducting surfaces facing each other must have equal and opposite charge densities. Here as the plate areas are equal,
$Q_2=-Q_3$.
Now PD inside the capacitor is given by
$= \frac {Q_2}{C} = \frac {2Q_2}{C} = \frac { Q_2 + Q_2}{2C}= \frac {Q_2 -Q_3}{2C}$
Hence (c) is the correct option

Question 12
A parallel plate capacitor of capacitance C is connected to a battery and is charged to a potential difference V. Another capacitor of capacitance 2C is connected to another battery and is charged to potential difference 2V. The charging batteries are now disconnected and the capacitors are connected in parallel to each other in such a way that the positive terminal of one is connected to the negative terminal of the other. The final energy of the configuration is
(a) $\frac {3CV^2}{2}$
(b) $\frac {25CV^2}{2}$
(c) $\frac {9CV^2}{2}$
(d) zero

Total charge =$(2C)(2V)+(C)(-V)=3CV$
Common potential =3CV/3C=V
Energy =$\frac {1}{2}(3C)(V)^2=\frac {3CV^2}{2}$
Hence (a) is the correct option

Question 13
A capacitor of 4 μ F is connected as shown in the circuit . The internal resistance of the battery is 0.5 Ω . The amount of charge on the capacitor plates will be

(a) 0
(b) 4 μ C
(c) 16 μ C
(d) 8 μ C

Potential difference across lower and middle branch of circuit is equal to the potential difference across capacitor of upper branch of circuit.
Current flows through 2 Ω resistance from left to right, is given by
$I=\frac {E}{R+r}=\frac {2.5}{2+ .5}=1A$
Therefore
The potential difference across 2Ω resistance
$V=IR= 1 \times 2 = 2V$
Hence potential difference across capacitor is also 2 V.
The charge on capacitor is $q = CV= (2 ) \times 2 V = 8$ μ C.

Question 14
Two condensers of capacities 2C and C are joined in parallel and charged upto potential V. The battery is removed and the condenser of capacity C is filled completely with a medium of dielectric constant K. The p.d. across the capacitors will now be
(a) $\frac {3V}{K+2}$
(b) $\frac {3V}{K}$
(c) $\frac {V}{K+2}$
(d) $\frac {V}{K}$

Initially
$C_1=2C$ and $C_2=C$
$q_1=2CV$
$q_2=CV$
Now condenser of capacity C is filled with dielectric K, therefore
$C'_2 = KC$
As charge is conserved
$q_1+q_2=(C'_2 + 2C)V'$
$V'=\frac {3V}{K+2}$
Hence (a) is correct

Question 15
Five identical capacitor plates each of area A are arranged such that adjacent plates are at a distance d apart . The plates are connected to a potential difference of EMF V as shown in the figure.

The charge on plates 1 and 4 will be
(a) $\frac {\epsilon _0 A V}{d}$,$\frac {2\epsilon _0 A V}{d}$
(b) $\frac {\epsilon _0 A V}{d}$,$\frac {-2\epsilon _0 A V}{d}$
(a) $\frac {-\epsilon _0 A V}{d}$,$\frac {2\epsilon _0 A V}{d}$
(a) $\frac {\epsilon _0 A V}{d}$,$\frac {-\epsilon _0 A V}{d}$

The charge on the plates are shown below

The arrangement is equivalent to four capacitors each of capacity
$C=\frac {\epsilon _0 A}{d}$
So charge in each capacitor is
$q= CV =\frac {\epsilon _0 A V}{d}$
Now for plate 1 is common to one capacitor and connected to positive terminal,so charge will be $\frac {\epsilon _0 A V}{d}$
Plate 4 is common to two capacitor and connected to negative terminal,so charge will be $\frac {-2\epsilon _0 A V}{d}$
Hence (b) is the correct option

Question 16
A parallel plate capacitor is connected to a battery as shown below

Consider two situations:
A: Key K is kept closed and plates of capacitors are moved apart using insulating handle.
B: Key K is opened and plates of capacitors are moved apart using insulating handle.
Choose the correct option(s).
(a) In A : Q remains same but C changes.
(b) In B : V remains same but C changes.
(c) In A : V remains same and hence Q changes.
(d) In B : Q remains same and hence V changes.

In case of A, Potential remains same as connected to battery, Now C changes as the distance change ,so the charge changes
In case of B, charge remains same as not connected to battery, Now C changes as the distance change ,so the potential difference changes
(c) and (d)