Equilibrium of Charges
Category : JEE Main & Advanced
(1) Definition : A charge is said to be in equilibrium, if net force acting on it is zero. A system of charges is said to be in equilibrium if each charge is separately in equilibrium.
(2) Type of equilibrium : Equilibrium can be divided in following type:
(i) Stable equilibrium : After displacing a charged particle from it's equilibrium position, if it returns back then it is said to be in stable equilibrium. If U is the potential energy then in case of stable equilibrium \[\frac{{{d}^{2}}U}{d{{x}^{2}}}\] is positive i.e., U is minimum.
(ii) Unstable equilibrium : After displacing a charged particle from it's equilibrium position, if it never returns back then it is said to be in unstable equilibrium and in unstable equilibrium \[\frac{{{d}^{2}}U}{d{{x}^{2}}}\] is negative i.e., U is maximum.
(iii) Neutral equilibrium : After displacing a charged particle from it's equilibrium position if it neither comes back, nor moves away but remains in the position in which it was kept it is said to be in neutral equilibrium and in neutral equilibrium \[\frac{{{d}^{2}}U}{d{{x}^{2}}}\] is zero i.e., U is constant
Different cases of equilibrium of charge
| Suspended charge | System of three collinear charge |
|
Freely suspended charge
In equilibrium \[QE=mg\] \[\Rightarrow E=\frac{mg}{Q}\] Suspension of charge from string
In equilibrium \[T\sin \theta =QE\] ...(i) \[T\cos \theta =mg\] ...(ii) From equations (i) and (ii) \[T=\sqrt{{{\left( QE \right)}^{2}}+{{\left( mg \right)}^{2}}}\] and \[\tan \theta =\frac{QE}{mg}\] |
In the following figure three charges \[{{Q}_{1}},\,Q\] and \[{{Q}_{2}}\] are kept along a straight line, charge Q will be in equilibrium if and only if |Force applied by charge \[{{Q}_{1}}\]| = |Force applied by charge \[{{Q}_{2}}\]|
i.e. \[\frac{{{Q}_{1}}Q}{x_{1}^{2}}=\frac{{{Q}_{2}}Q}{x_{2}^{2}}\] \[\Rightarrow \frac{{{Q}_{1}}}{{{Q}_{2}}}={{\left( \frac{{{x}_{1}}}{{{x}_{2}}} \right)}^{2}}\] This is the necessary condition for Q to be in equilibrium. If all the three charges (\[{{Q}_{1}},\,Q\] and \[{{Q}_{2}}\]) are similar, Q will be in stable equilibrium. If extreme charges are similar while charge Q is of different nature so Q will be in unstable equilibrium. |
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