Capacitance Formulas

1. Capacitance of a conductor

Capacity of storing charge
C = $\frac{Q}{V}$
Unit → farad = $\frac{\text{ coulomb }}{\text{ volt }}$

2. Capacitance of a spherical conductor

C = 4πε₀R
R → Radius of conductor.

3. Common potential when two charged conductors are connected

C = C₁ + C₂
Q = Q₁ + Q₂ = C₁V₁ + C₂ V₂
Common potential
V = $\frac{{\mathrm{C}}_1{\mathrm{V}}_1+{\mathrm{C}}_2{\mathrm{V}}_2}{{\mathrm{C}}_1+{\mathrm{C}}_2}$
Charge transferred
ΔQ = $\frac{C_1{C}_2}{C_1+C_2}$ (V₁ – V₂)
Energy loss
ΔU = $\frac{1}{2}\frac{C_1{C}_2}{C_1+C_2}$ (V₁ – V₂)²

4. Capacitor (condenser)

An arrangement of conductors for increasing the capacitance. It has two conductors placed nearby, one is charged and the other is earthed.

5. Parallel plate capacitor

C = $\frac{{ϵ}_{0}A}{d}$, C_m = ε_r = $\left(\frac{{ϵ}_{0}A}{d}\right)$ = ε_rC
If a dielectric of thickness t is placed in between, then
C = $\frac{{ϵ}_{0}A}{(d-t+\frac{t}{{ϵ}_r})}$
(a)

If slabs of thickness t₁, t₂, t₃ …. t_n of dielectric constant’s ε_r1, ε_r2, …… ε_rn are placed in between
C = $\frac{{ϵ}_{0}A}{(\frac{t_1}{{ϵ}_{r_1}}+\frac{t_2}{{ϵ}_{r_2}}+\dots .+\frac{t_n}{{ϵ}_{r_n}})}$
and d = t₁ + t₂ + t₃ + …. + t_n
(b)

6. Spherical capacitor

When outer spherical shell is earthed
C = 4πε₀ε_r $\frac{{\mathrm{R}}_1{\mathrm{R}}_2}{{\mathrm{R}}_2-{\mathrm{R}}_1}$ ; R₂ > R₁
When inner sphere is earthed
C = 4πε₀ε_r $\frac{{\mathrm{R}}_1{\mathrm{R}}_2}{{\mathrm{R}}_2-{\mathrm{R}}_1}$ + 4πε₀R₂

7. Cylinderical condenser

C = $\frac{2\pi {ϵ}_0{ϵ}_{\mathrm{r}}\ell }{{\log }_{\mathrm{c}}\left({\mathrm{R}}_2/{\mathrm{R}}_1\right)}$

8. Capacitance of a two wire line

C = $\frac{\pi {ϵ}_0{ϵ}_1\ell }{{\log }_e(d/r)}$ (d >> r)
l → length of each wire
d → distance between wires
r → radius of each wires

9. Multiplate capacitor

C = (n – 1) $\frac{{ϵ}_0{\in }_{r}A}{d}$
A → area of each plate.

10. Energy stored in a capacitor

U = $\frac{1}{2}$ CV² = $\frac{1}{2}$ QV = $\frac{1}{2}\frac{Q^2}{C}$
This energy resides in electric field. Energy density of electric field
U = $\frac{1}{2}$ εE² = $\frac{\text{ Total Energy }}{\text{ Volume }}$

11. Combination of capacitor

Series combination
$\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+\dots \dots +\frac{1}{C_n}$
Q₁ = Q₂ = …….. = Q
V = V₁ + V₂ + V₃+ ……… + V_n
Parallel combination
C = C₁ + C₂ + …….. C_n
V₁ = V₂ = …….. = V
Q = Q₁ + Q₂ + Q₃ + …….. + Q_n

12. Charging and discharging of a capacitor through a resistance

Charging
q = q₀ (1 – e^-t/RC)
V = V₀ (1 – e^-t/RC)

I = I₀ e^-t/RC, I₀ = $\frac{V_0}{R}$
V → P.d. across capacitor
q → charge on the capacitor
I → current through the capacitor

Discharging
q = q₀ e^-t/RC
V = V₀ e^-t/RC
I = – I₀ e^-t/RC

Time constant
while charging
τ = RC, time in which q = 0.63Q, V = 0.63V₀ and I = 0.37I₀
while discharging q = 0.37Q, V = 0.37V₀ and I = 0.37I₀

• In charging energy supplied by the battery up to steady state is V₀q₀
• Energy stored in capacitor as a electric field is $\frac{1}{2}$V₀q₀
• Energy loss during charges = $\frac{1}{2}$ V₀q₀

13. Force of attraction between the plates of a capacitor

F = $\frac{1}{2}$ εE² A = $\frac{Q^2}{2\in A}=\frac{1}{2}\frac{C{V}^2}{d}$
F = $\frac{{\sigma }^2}{2ϵ}$A

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