ELECTROSTATIC POTENTIAL AND CAPACITANCE

LEARNING SHEET-5

Dear students, in this worksheet we shall discuss

about an important component of an electric circuit –
Capacitor. Capacitor also known as Electric-
condenser is a two terminal electric component
which has ability or capacity to store energy in the
form of electric charge. Capacitors are usually
designed to enhance and increase the effect of
capacitance considering the capacitor’s size and
shape. The storing capacity of capacitance varies from
small storage to high storage.

READ & EXPLAIN

Before you proceed, click here for a small description about capacitors.

A capacitor is a system of two conductors separated by an insulator. The conductors have charges; say Q 1 and Q2, and
potentials V1 and V2. Usually, in practice, the two conductors have charges Q and – Q, with potential difference V = V1 –
V2 between them. We shall consider only this kind of charge configuration of the capacitor. (Even a single conductor can
be used as a capacitor by assuming the other at infinity.) The conductors may be so charged by connecting them to the
two terminals of a battery. Q is called the charge of the capacitor, though this, in fact, is the charge on one of the
conductors – the total charge of the capacitor is zero. The electric field in the region between the conductors is
proportional to the charge Q. That is, if the charge on the capacitor is, say doubled, the electric field will also be doubled
at every point. Now, potential difference V is the work done per unit positive charge in taking a small test charge from
the conductor 2 to 1 against the field. Consequently, V is also proportional to Q, and the ratio Q/V is a constant:
𝑄
𝐶=
𝑉
The constant C is called the capacitance of the capacitor. C is independent of Q or V, as stated above. The capacitance C
depends only on the geometrical configuration (shape, size, separation) of the system of two conductors. The SI unit of
capacitance is 1 farad (=1 coulomb volt-1) or 1 F = 1 C V–1. A capacitor with fixed capacitance is symbolically shown as
⟛, while the one with variable capacitance is shown as ⟛ . Equation shows that for large C, V is small for a given Q.
This means a capacitor with large capacitance can hold large amount of charge Q at a relatively small V. High potential
difference implies strong electric field around the conductors. A strong electric field can ionize the surrounding air and
accelerate the charges so produced to the oppositely charged plates, thereby neutralizing the charge on the capacitor
plates, at least partly. In other words, the charge of the capacitor leaks away due to the reduction in insulating power of
the intervening medium.
The maximum electric field that a dielectric medium can withstand without break-down (of its insulating property) is
called its dielectric strength; for air it is about 3 × 106 Vm–1. For a separation between conductors of the order of 1 cm or
so, this field corresponds to a potential difference of 3 × 104 V between the conductors. Thus, for a capacitor to store a
large amount of charge without leaking, its capacitance should be high enough so that the potential difference and
hence the electric field do not exceed the break-down limits. Put differently, there is a limit to the amount of charge that
can be stored on a given capacitor without significant leaking. In practice, a farad is a very big unit; the most common
units are its sub-multiples 1 µF = 10–6 F, 1 nF = 10–9 F, 1 pF = 10–12 F, etc.
1. Show graphically the effect on capacitance due to electric charge and voltage.
2. What is the dimensional formula of Capacitance?
3. A conductor is said to have a capacity of one farad, when a ______ raises its ______ by _______.
4. In what form is the energy stored in a charged capacitance?
5. Can we give as much charge to a capacitor as we wish?
6. What is the net charge on a charged capacitor?
7. What is the relationship between capacitance C and radius r for a spherical conductor?
Answer the following questions in the notebook:

8. The capacitance of a conductor is 1 Farad. What do you mean by this statement?

9. On which factors does the capacitance of a capacitor depend?
10. Write two applications of capacitors in electrical circuits.
11. Diameter of a spherical conductor is 1 metre. What is its capacity?
12. Eight identical spherical drops, each carrying charge 1 nC are at a potential of 900 V each. All these drops
combine together to form a single large drop. Calculate the potential of this large drop. Assume no wastage of
any kind. Take capacitance of a sphere of radius r as proportional to r.
13. An uncharged insulated conductor A is brought near a charged insulated conductor B. What happens to charge
and potential of B?

READ & EXPLAIN

Parallel Plate Capacitors are formed by an arrangement of electrodes and insulating material or dielectric. A parallel
plate capacitor can only store a finite amount of energy before dielectric breakdown occurs. It can be defined as:
When two parallel plates are connected across a battery, the plates are charged and an electric field is established
between them, and this setup is known as the parallel plate capacitor.
The figure below depicts a parallel plate capacitor. We can see two large
plates placed parallel to each other at a small distance d. The distance
between the plates is filled with a dielectric medium as shown by the
dotted array. The two plates carry an equal and opposite charge.
Here, we see that the first plate carries a charge +Q and the second carries
a charge –Q. The area of each of the plates is A and the distance between
these two plates is d. The distance d is much smaller than the area of the
plates and we can write d<<A, thus the effect of the plates are considered
as infinite plane sheets and the electric field generated by them is treated
as that equal to the electric field generated by an infinite plane sheet of
uniform surface charge density. As the total charge on plate 1 is Q and the area of the plate is A, the surface charge
density can be given as

Similarly, for plate 2 with a total charge equal to –Q and area A, the surface charge density can be given as,

We divide the regions around the parallel plate capacitor into three parts, with area 1 being the area left to the first
plate, area 2 being the area between the two planes and area 3 is the area to the right of plate 2.
Let us calculate the electric field in the region around a parallel plate capacitor.
Region I: The magnitude of the electric field due to both the infinite plane sheets I and II is the same at any point in this
region, but the direction is opposite to each other, the two forces cancel each other and the overall electric field can be
given as,

Region II: The magnitude, as well as the direction of the electric field due to both the plane sheets I and II in these
regions, is the same and the overall effect can be given as,

Region III: Similar to the region I, here too, the magnitude of the electric field generated due to both the plane sheets I
and II is the same but the direction is opposite, giving the same result as,
Here, the electric field is uniform throughout and its direction is from the positive plate to the negative plate.
The potential difference across the capacitor can be calculated by multiplying the electric field and the distance between
the planes, given as,

And the capacitance for the parallel plate capacitor can be given as,

Where,
ϵo is the permittivity of space (8.54 × 10−12 F/m), k is the relative permittivity of dielectric material, d is the separation
between the plates and A is the area of plates
For typical values like A = 1 m2 , d = 1 mm, we get
8.85𝑋10−12 𝐶 2 𝑁−1 𝑚−2 𝑋1𝑚2
𝐶=
10−3 𝑚
(You can check that if 1F= 1C.V–1 = 1C (NC–1m)–1 = 1 C2 N–1m–1.) This shows that 1F is too big a unit in practice, as
remarked earlier.
The direction of the electric field is defined as the direction in which the positive test charge would flow. Capacitance is
the limitation of the body to store the electric charge. Every capacitor has its capacitance. The typical parallel-plate
capacitor consists of two metallic plates of area A, separated by the distance d.
14. Calculate the area of the plates needed to have C = 1F for a separation of, say 1 cm.
15. What is the effect of presence of a dielectric medium on (a) capacitance of a parallel plate capacitor? (b)
electrostatic force between two charges?
16. The distance between the plates of a parallel plate capacitor is d. A metal plate of thickness d/2 is placed
between the plates, what will be the new capacity?
17. Why is earth considered as zero of potential in practice? Justify.

Answer the following questions in the notebook:

18. A parallel plate capacitor with air between the plates has a capacitance of 8 pF (1pF = 10–12 F). What will be the
capacitance if the distance between the plates is reduced by half, and the space between them is filled with a
substance of dielectric constant 6?
19. In a parallel plate capacitor with air between the plates, each plate has an area of 6 × 10–3 m2 and the distance
between the plates is 3 mm. Calculate the capacitance of the capacitor. If this capacitor is connected to a 100 V
supply, what is the charge on each plate of the capacitor?
20. A 12pF capacitor is connected to a 50V battery. How much electrostatic energy is stored in the capacitor?
21. A 600pF capacitor is charged by a 200V supply. It is then disconnected from the supply and is connected to
another uncharged 600 pF capacitor. How much electrostatic energy is lost in the process?
22. The plates of a parallel plate capacitor have an area of 90 cm2 each and are separated by 2.5 mm. The capacitor
is charged by connecting it to a 400 V supply. (a) How much electrostatic energy is stored by the capacitor? (b)
View this energy as stored in the electrostatic field between the plates, and obtain the energy per unit volume u.
Hence arrive at a relation between u and the magnitude of electric field E between the plates.

Click here to find how a capacitor charges and discharges itself.