Capacitor is a device that stores electrostatic field energy. Capacitor provides temporary storage of energy in circuits and release the stored energy when required. Capacitance is the measure of energy that a capacitor can store.
Capacitance
Potential of an isolated conductor is directly proportional to charge present on it .
V α Q
Inserting constant of proportionality
Q = C x V
or C = Q / V
Q / V is constant for isolated conductor and is defined as Capacitance of conductor .
If we assume, V = 1 volt
then C = Q
Hence , Capacitance of an isolated conductor is charge required to rise the potential of isolated conductor by 1 volt .
Units of capacitance –
SI unit for capacitance is Farad .
1 microfarad(1μF) = 10⁻⁶ Farad
1 picofarad (1pF) = 10⁻¹² Farad
Spherical Capacitor and its Capacitance
Spherical capacitor consists of a solid or a hollow spherical conductor surrounded by another concentric hollow spherical conductor. The outer shell is earthed .
Suppose a charge +Q is present on the inner shell. Due to +Q charge on inner shell, there is charge – Q (induced) on the inner surface of the outer shell and + Q on the outer surface of the outer shell (as shown in following figure). Since, outer shell is earthed that’s why charge +Q will flow from the outer shell to the Earth.
Capacitance ( C ) = Q / potential difference
As we know outer shell is earthed that’s why potential on outer shell is zero.
Potential on inner shell = ( Q/4πε₀ ) [1/b – 1/a]
⇒ C = Q/ V₂= 4πε₀ab/(b-a)
Cylindrical Capacitor and its Capacitance
Cylindrical capacitor consist of a solid or hollow cylindrical conductor surrounded by another coaxial hollow cylindrical conductor .
Potential difference = (q / 2πε₀L) ln(r₂/r₁)
⇒ Capacitance = 2πε₀L/ ln (r₂/r₁)
Parallel Plate Capacitor and its Capacitance
A parallel plate capacitor consists of two large plates placed parallel to each other with a separation d. d is very small in comparison to the length and breadth of plates.
We know that electric field near a very large sheet is given by σ /2ε₀
Thus, Electric field between the plates is,
E = σ/2ε₀ + σ/2ε₀ = σ/ε₀ = Q/Aε₀ {σ = Q/A}
Potential difference between the plates is,
V = Ed = Qd/Aε₀
⇒ C = Q /V = ε₀A / d
Factors affecting Capacitance –
Capacitance is function of physical dimensions of the conductor and permittivity of the dielectric present inside it. So, Capacitance mainly depends upon area of plate , distance between plate and dielectric present inside it
- Area –
Capacitance of parallel plate capacitor is proportional to area of plates. All other factors being equal , greater plat area gives grater capacitance and less plate area gives less capacitance.
- Distance between plates-
Capacitance of parallel plate capacitor is inversely proportional to distance between plates. All the factors being equal further plate spacing gives less capacitance, closer plate spacing gives more capacitance.
- Capacity of Parallel Plate capacitor with dielectric –
Suppose a parallel plate capacitor with area A and a separation d. A dielectric slab of thickness t, area A and dielectric constant K is inserted between the plates.
If Q charge is given to the capacitor plates then,
Electric field between the plates of the capacitor is outside the slab –
E₀ = σ / ε₀ = Q / Aε₀
Electric field inside dielectric slab –
E= E₀/ K = Q / Kε₀A
Potential difference between plate –
V = V₁ + V₂ + V₃
⇒ Q / C = E₀ t₁ + E t₂ + E₀ t₃
⇒ C = [ε₀ A / (d-t)] + t/k
Special Cases :
- for conducting slab (metal plate) –
Dielectric constant (K) = ∞ (infinity)
⇒ t / K = 0
⇒ C = ε₀A /(d-t)
- for conducting slab with t <<< d –
C = ε₀ A / d
Series Combination of Capacitors
Suppose three capacitors with capacitance C₁, C₂ and C₃ are connected in series with a battery of voltage V as shown in figure.
In series combination V = V₁ + V₂ + V₃
⇒ Q / C = Q / C₁ + Q/ C₂ + Q/ C₃
Thus, 1/ C = 1/C₁ + 1/C₂ + 1/C₃
- In series combination equivalent capacitance is always less than any of the individual capacitance.
- Charge on capacitors are same but potential across the capacitors are different.
Parallel Combination of Capacitors
Suppose three capacitor with capacitance C₁, C₂ and C₃ are connected in parallel with a battery of voltage V as shown in fig.
In parallel combination q = q₁ + q₂ + q₃
⇒ CV = C₁V + C₂V + C₃V
Thus, C = C₁ + C₂ + C₃
- In parallel combination equivalent capacitance is always greater than any of the individual capacitance.
- Charge on capacitors are different but potential across the capacitors are same.
Sharing of Charges
When two conductors are connected by a connecting wire, charge flows from one conductor to another (higher potential to lower potential). This charge flow phenomenon takes place until both the conductor have same potential.
Q₁+Q₂=Q₁´+Q₂´ =Q
Q₂´= Q[r₂/ (r₂+ r₁)]
Q₁´= Q[r₁/ (r₁+r₂)]
Common potential,V = C₁V₁+C₂V₂/(C₁+C₂)
Therefore, Energy loss =[ C₁C₂/2(C₁+C₂)] (V₁-V₂)²