JEE Main & Advanced Physics Electrostatics & Capacitance Time Period of Oscillation of a Charged Body

 (1) Simple pendulum based : If a simple pendulum having length l and mass of bob m oscillates about it's mean position than it's time period of oscillation \[T=2\pi \sqrt{\frac{l}{g}}\]

Case-1 : If some charge say +Q is given to bob and an electric field E is applied in the direction as shown in figure then equilibrium position of charged bob (point charge) changes from O to O'.

On displacing the bob from it's equilibrium position 0 it will oscillate under the effective acceleration g', where

\[mg'=\sqrt{{{\left( mg \right)}^{2}}+{{\left( QE \right)}^{2}}}\] \[\Rightarrow g'=\sqrt{{{g}^{2}}+{{\left( QE/m \right)}^{2}}}\]. 

Hence the new time period is  \[{{T}_{1}}=2\pi \sqrt{\frac{l}{g'}}\]\[=2\pi \sqrt{\frac{l}{\left( {{g}^{2}}+\left( QE/m \right){{\,}^{2}} \right){{\,}^{\frac{1}{2}}}}}\]

Since \[g'>g,\] so \[{{T}_{1}}<T\] i.e. time period of pendulum will decrease.

Case-2 : If electric field is applied in the downward direction then. Effective acceleration

\[g'=g+QE/m\]

So new time period

\[{{T}_{2}}=2\pi \sqrt{\frac{l}{g+\left( QE/m \right)}}\]

\[{{T}_{2}}<T\]

Case-3 : In case 2 if electric field is applied in upward direction then, effective acceleration.

\[g'=g-QE/m\]

So new time period

\[{{T}_{3}}=2\pi \sqrt{\frac{l}{g-\left( QE/m \right)}}\]

\[{{T}_{3}}>T\]

(2) Charged circular ring : A thin stationary ring of radius R has a positive charge +Q unit. If a negative charge \[-q\] (mass m) is placed at a small distance \[x\] from the centre. Then motion of the particle will be simple harmonic motion.

Having time period \[T=2\pi \sqrt{\frac{4\pi {{\varepsilon }_{0}}m{{R}^{3}}}{Q\,q}}\]

(3) Spring mass system : A block of mass m containing a negative charge \[-Q\] is placed on a frictionless horizontal table and is connected to a wall through an unstretched spring of spring constant k as shown. If electric field E applied as shown in figure the block experiences an electric force, hence spring compress and block comes in new position. This is called the equilibrium position of block under the influence of electric field. If block compressed further or stretched, it execute oscillation having time period 

\[T=2\pi \sqrt{\frac{m}{k}}\].

Maximum compression in the spring due to electric field \[=\frac{QE}{k}\]