Capacitor in Series and Parallel

We know that in the series combination current will be same across all the capacitors and voltage gets divided across them

i = c dv/dt ⇒ V = ∫ (i/c) dt

v = v₁ + v₂ + v₃

∫ (i_eq/c_eq) dt = ∫ (i₁/c₁) dt + ∫ (i₂/c₂) dt + … + ∫ (i_n/c_n) dt

i = i₁ = i₂ = i_n

∫ (i/c_eq) dt = ∫ (i/c₁) dt + ∫ (i/c₂) dt + … + ∫ (i/c_n) dt

i/c_eq = i/c₁ + i/c₂ + … + i/c_n

C_eq = 1/ (1/c₁ + 1/c₂ + 1/c_n)

In the parallel combination voltage will be same across all the capacitors and current gets divided across all the capacitors

i = i₁ + i₂ + i₃

i = c dv/dt

c_eq dv/dt = c₁ dv₁/dt + c₂ dv₂/dt + … + c_n d v_n /dt

v = v₁ = v₂ = … = v_n

c_eq dv/dt = c₁ dv/dt + c₂ dv/dt + … + c_n dv/dt

c_eq = c₁ + c₂ + … + c_n