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.

(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)

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)

        (c) On the surface, V_surface = (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]

        (c) On the surface, V_surface = (1/4π ε₀) (q/R)

       (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     

       (c) q₁/q₂ = r₁/r₂ or σ₁/ σ₂ = 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

     (c) V = n^2/3V_small

    (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:-

     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 q₁, 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.

      (a) If θ = 90º, then W = 0

      (b) If θ = 0º, then W = -pE

      (c) If θ = 180º, 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:-

        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 ε₀b

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_e b/a)] = [2π ε₀l /(log_e b/a) ]

       C_med = [2πKε₀l /(log_e b/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

           _?

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

?

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.

       (ii) q₁ = q₂ = q₃ = q

      (iii) V₁= q/C₁, V₂= q/C₂, V₃= q/C₃

     (iv) Energy Stored, U = U₁+U₂+U₃

• Energy stored in a group of capacitors:-

      (a) Energy stored in a series combination of capacitors:-

        W = ½ (q²/C₁) + ½ (q²/C₂) + ½ (q²/C₃) = W₁+W₂+W₃

Thus, net energy stored in the combination is equal to the sum of the energies stored in the component capacitors.

      (b) Energy stored in a parallel combination of capacitors:-

       W = ½ C₁V ² +½ C₂V ² + ½ C₃V ² = W₁+W₂+W₃

The net energy stored in the combination is equal to sum of energies stored in the component capacitors.

• Force of attraction between plates of a charged capacitor:-

      (a) F = ½ ε₀E²A

      (b) F = σ²A/2ε₀

      (c) F=Q²/2ε₀A

• Force on a dielectric in a capacitor:-

       F = (Q²/2C²) (dC/dx) = ½ V² (dC/dx)

• Common potential when two capacitors are connected:-

        V = [C₁V₁+ C₂V₂] / [C₁+C₂] = [Q₁+Q₂]/ [C₁+C₂]

• Charge transfer when two capacitors are connected:-

       ΔQ = [C₁C₂/C₁+C₂] [V₁-V₂]

• Energy loss when two capacitors are connected:-

         ΔU = ½ [C₁C₂/C₁+C₂] [V₁-V₂] ²

• Charging of a capacitor:-

         (a) Q = Q₀(1-e^-t/RC)

         (b) V = V₀(1-e^-t/RC)

         (c) I = I₀(1-e^-t/RC)

         (d) I₀ = V₀/R

• Discharging of a capacitor:-

        (a) Q = Q₀(e^-t/RC)

        (b) V = V₀(e^-t/RC)

         (c)  I = I₀(e^-t/RC)

• Time constant:-