Advertisements
Advertisements
Questions
Obtain the expression for capacitance for a parallel plate capacitor.
Derive an expression for the capacitance of a parallel plate capacitor with air present between the two plates.
Advertisements
Solution 1
Consider a capacitor with two parallel plates each of cross-sectional area A and separated by a distance d. The electric field between two infinite parallel plates is uniform and is given by E = `sigma/ε_0` where σ is the surface charge density on the plates σ = `"Q"/"A"`. If the separation distance d is very much smaller than the size of the plate (d2 << A), then the above result is used even for finite-sized
parallel plate capacitor.
Capacitance of a parallel plate capacitor
Capacitance of a parallel plate capacitor
The electric field between the plates is
E = `"Q"/("A"ε_0)` ....(1)
Since the electric field is unifonn, the electric potential between the plates having separation d is given by
V = Ed = `"Qd"/("A"ε_0)` ....(2)
Therefore the capacitance of the capacitor is given by
C = `"Q"/"V" = "Q"/(("Qd"/("A"ε_0))) = (ε_0"A")/"d"` ....(3)
From equation (3), it is evident that capacitance is directly proportional to the area of cross-section and is inversely proportional to the distance between the plates. This can be understood from the following.
- If the area of cross-section of the capacitor plates is increased, more charges can be distributed for the same potential difference. As a result, the capacitance is increased.
- If the distance d between the two plates is reduced, the potential difference between the plates (V = Ed) decreases with E constant.
Solution 2
Derivation of the expression for the capacitance.
Let the two plates be kept parallel to each other separated by a distance d and the cross-sectional area of each plate is A. Electric field by a single thin plate E = `sigma/(2epsi_0)`
The total electric field between the plates E = `sigma/epsi_0` = `Q/(Aepsi_0)`
The potential difference between the plates V = Ed = `[Q/(Aε_0)]`d.
Capacitance C = `"Q"/"V" = (Aepsi_0)/d`
RELATED QUESTIONS
When an AC source is connected to an ideal capacitor, show that the average power supplied by the source over a complete cycle is zero
A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed and the inner sphere is given a charge of 2.5 µC. The space between the concentric spheres is filled with a liquid of dielectric constant 32.
(a) Determine the capacitance of the capacitor.
(b) What is the potential of the inner sphere?
(c) Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm. Explain why the latter is much smaller.
Two identical capacitors of 12 pF each are connected in series across a battery of 50 V. How much electrostatic energy is stored in the combination? If these were connected in parallel across the same battery, how much energy will be stored in the combination now?
Also find the charge drawn from the battery in each case.
A capacitor of unknown capacitance is connected across a battery of V volts. The charge stored in it is 360 μC. When potential across the capacitor is reduced by 120 V, the charge stored in it becomes 120 μC.
Calculate:
(i) The potential V and the unknown capacitance C.
(ii) What will be the charge stored in the capacitor, if the voltage applied had increased by 120 V?
A capacitor of capacitance ‘C’ is being charged by connecting it across a dc source along with an ammeter. Will the ammeter show a momentary deflection during the process of charging? If so, how would you explain this momentary deflection and the resulting continuity of current in the circuit? Write the expression for the current inside the capacitor.
Three identical capacitors C1, C2 and C3 of capacitance 6 μF each are connected to a 12 V battery as shown.
Find
(i) charge on each capacitor
(ii) equivalent capacitance of the network
(iii) energy stored in the network of capacitors
A thin metal plate P is inserted between the plates of a parallel-plate capacitor of capacitance C in such a way that its edges touch the two plates . The capacitance now becomes _________ .
The capacitance of a capacitor does not depend on
The plates of a parallel-plate capacitor are made of circular discs of radii 5⋅0 cm each. If the separation between the plates is 1⋅0 mm, what is the capacitance?
Each of the plates shown in figure has surface area `(96/∈_0) xx 10^-12` Fm on one side and the separation between the consecutive plates is 4⋅0 mm. The emf of the battery connected is 10 volts. Find the magnitude of the charge supplied by the battery to each of the plates connected to it.
When air is replaced by a dielectric medium of constant K, the maximum force of attraction between two charges separated by a distance ______.
The capacitance of a parallel plate capacitor is 60 µF. If the distance between the plates is tripled and area doubled then new capacitance will be ______.
The radius of a sphere of capacity 1 microfarad in the air is ______
Two similar conducting spheres having charge+ q and -q are placed at 'd' seperation from each other in air. The radius of each ball is r and the separation between their centre is d (d >> r). Calculate the capacitance of the two ball system ______.
A 5µF capacitor is charged fully by a 220 V supply. It is then disconnected from the supply and is connected in series to another uncharged 2.5 µF capacitor If the energy change during the charge redistribution is `"X"/100`J then value of X to the 100 nearest integer is ______.
Current versus time and voltage versus time graphs of a circuit element are shown in figure.
 |
 |
The type of the circuit element is ______.
The plates of a parallel plate capacitor are separated by d. Two slabs of different dielectric constant K1 and K2 with thickness `3/8 d and d/2`, respectively, are inserted in the capacitor. Due to this, the capacitance becomes two times larger than when there is nothing between the plates. If K1 = 1.25 K, the value of K1 is:
If the plates of a parallel plate capacitor connected to a battery are moved close to each other, then:
- the charge stored in it. increases.
- the energy stored in it, decreases.
- its capacitance increases.
- the ratio of charge to its potential remains the same.
- the product of charge and voltage increases.
Choose the most appropriate answer from the options given below:
Two spherical conductors of capacities 4 µF and 3 µF are charged to same potential having radii 4 cm and 3 cm respectively. If ‘σ1’ and ‘σ2’ represent surface density of charge on respective conductors then ______.
