Revision Notes on Electrostatic Potential and Capacitance

Electric Potential:-

(a) Electric potential, at any point, is defined as the negative line integral of electric field from infinity to that point along any path.

$V (\vec{r}) = -\int_{\infty }^{r}\vec{E}.d\vec{r}$

(b) V(r) = kq/r

(c) Potential difference, between any two points, in an electric field is defined as the work done in taking a unit positive charge from one point to the other against the electric field.

W_AB = q [V_A-V_B]

So, V = [V_A-V_B] = W/q

Units:- volt (S.I), stat-volt (C.G.S) Electric Potential

Dimension:- [V] = [ML²T^-3A^-1]

Relation between volt and stat-volt:- 1 volt = (1/300) stat-volt

Relation between electric field (E) and electric potential (V):-

E = -dV/dx = --dV/dr

Potential due to a point charge:-

V = (1/4π ε₀) (q/r)

Potential at point due to several charges:-

V = (1/4π ε₀) [q₁/r₁ + q₂/r₂ + q₃/r₃]

= V₁+V₂+ V₂+….

Potential due to charged spherical shell:-

(a) Outside, V_out = (1/4π ε₀) (q/r)

(b) Inside, V_in = - (1/4π ε₀) (q/R)

Potential due to a uniformly charged non-conducting sphere:-

(a) Outside, V_out = (1/4π ε₀) (q/r)

(b) Inside, V_in = (1/4π ε₀) [q(3R²-r²)/2R³]

(d) In center, V_center = (3/2) [(1/4π ε₀) (q/R)] = 3/2 [V_surface]

Common potential (two spheres joined by thin wire):-

(a) Common potential, V = (1/4π ε₀) [(Q₁+Q₂)/(r₁+r₂)]

(b) q₁ = r₁(Q₁+Q₂)/(r₁+r₂) = r₁Q/ r₁+r₂ ; q₂ = r₂Q/ r₁+r₂

Potential at any point due to an electric dipole:-

V (r,θ) = qa cosθ/4πε₀r² = p cosθ/4πε₀r²

(a) Point lying on the axial line:- V = p/4πε₀r²

(b) Point situated on equatorial lines:- V = 0

If n drops coalesce to form one drop, then,

(a) R = n^1/3r

(b) Q = nq

(d) σ = n^1/3 σ_small

(e) E = n^1/3 E_small

Electric potential energy U or work done of the system W having charge q₁ and q₂:-

Electric Potential Energy

W = U = (1/4πε₀) (q₁q₂/r₁₂) = q₁V₁

Electric potential energy U or work done of the system W of a three particle system having charge q_1, q₂ and q₃:-

W = U = (1/4πε₀) (q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₂q₃/r₂₃)

Electric potential energy of an electric dipole in an electric field:- Potential energy of an electric dipole, in an electrostatic field, is defined as the work done in rotating the dipole from zero energy position to the desired position in the electric field.

? $W = - \vec{p}.\vec{E} = - pE cos\theta$

(a) If θ = 90º, then W = 0

(b) If θ = 0º, then W = -pE

Kinetic energy of a charged particle moving through a potential difference:-

K. E = ½ mv²= eV

Conductors:- Conductors are those substance through which electric charge easily.
Insulators:- Insulators (also called dielectrics) are those substances through which electric charge cannot pass easily.
Capacity:- The capacity of a conductor is defined as the ratio between the charge of the conductor to its potential

C = Q/V

Units:-

S.I – farad (coulomb/volt)

C.G.S – stat farad (stat-coulomb/stat-volt)

Dimension of C:- [M^-1L^-2T⁴A²]

Capacity of an isolated spherical conductor:-

C = 4πε₀r

Capacitor:- A capacitor or a condenser is an arrangement which provides a larger capacity in a smaller space.
Capacity of a parallel plate capacitor:-

Capacity of a Parallel Plate Capacitor

C_air= ε₀A/d

C_med = Kε₀A/d

Here, A is the common area of the two plates and d is the distance between the plates.

Effect of dielectric on the capacitance of a capacitor:-

C = ε₀A/[d-t+(t/K)]

Here d is the separation between the plates, t is the thickness of the dielectric slab A is the area and K is the dielectric constant of the material of the slab.

If the space is completely filled with dielectric medium (t=d), then,

C = ε₀KA/ d

Capacitance of a sphere:-

(a) C_air = 4πε₀R

(b) C_med = K (4πε₀R)

Capacity of a spherical condenser:-

(a) When outer sphere is earthed:-

C_air = 4πε₀ [ab/(b-a)]

C_med = 4πε₀ [Kab/(b-a)]

(b) When the inner sphere is earthed:-

C₁= 4πε₀ [ab/(b-a)]

C₂ = 4πε₀b?

Net Capacity, C '=4πε₀[b²/b-a]

Increase in capacity, ΔC = 4π ε₀

It signifies, by connecting the inner sphere to earth and charging the outer one we get an additional capacity equal to the capacity of outer sphere.

Capacity of a cylindrical condenser:-

C_air = λl / [(λ/2π ε₀) (log_eb/a)] = [2π ε₀l /(log_eb/a) ]

C_med = [2πKε₀l /(log_eb/a) ]

Potential energy of a charged capacitor (Energy stored in a capacitor):-

W = ½ QV = ½ Q²/C = ½ CV²

Energy density of a capacitor:-

U = ½ ε₀E² = ½ (σ²/ ε₀)

This signifies the energy density of a capacitor is independent of the area of plates of distance between them so long the value of E does not change.

Grouping of Capacitors:-

(a)

(i) Capacitors in parallel:- C = C₁+C₂+C₃+…..+C_n ?

Three Capacitors C1, C2 and C3 are in Parallel Connection. The resultant capacity of a number of capacitors, connected in parallel, is equal to the sum of their individual capacities.

(ii)V₁= V₂= V₃ = V

(iii) q₁ =C₁V, q₂ = C₂V, q₃ = C₃V

(iv) Energy Stored, U = U₁+U₂+U₃

(b)

(i) Capacitors in Series:- 1/C = 1/C₁+ 1/ C₂+……+ 1/C_n

Three Capacitors C1, C2 and C3 are in Series Connection. ?The reciprocal of the resultant capacity of a number of capacitors, connected in series, is equal to the sum of the reciprocals of their individual capacities.