question_answer

A charge q is placed at the centre of the line joining two equal charges Q. The system of the three charges will be in equilibrium, if \[q\] is equal to             [IIT 1987; CBSE PMT 1995; Bihar MEE 1995; CPMT 1999; MP PET 1999; MP PMT 1999, 2000; RPET 1999;  KCET 2001; AIEEE 2002; AFMC 2002; Kerala PMT 2004; J & K CET 2004]

A)                                                      \[-\frac{Q}{2}\]                   

B)            \[-\frac{Q}{4}\]

C)                    \[+\frac{Q}{4}\]

D)                                      \[+\frac{Q}{2}\]

Correct Answer: B

Solution :

             Suppose in the following figure, equilibrium of charge B is considered. Hence for it's equilibrium \[|{{F}_{A}}|\,=\,|{{F}_{C}}|\] Þ \[\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{Q}^{2}}}{4{{x}^{2}}}=\]\[\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{qQ}{{{x}^{2}}}\] Þ \[q=\frac{-\,Q}{4}\]   Short Trick: For such type of problem the magnitude of middle charge can be determined if either of the extreme charge is in equilibrium by using the following formula. If charge A is in equilibrium then q = ? \[{{Q}_{B}}\,{{\left( \frac{{{x}_{1}}}{x} \right)}^{2}}\] If charge B is in equilibrium then \[q=-{{Q}_{A}}\,\,{{\left( \frac{{{x}_{2}}}{x} \right)}^{2}}\] If the whole system is in equilibrium then use either of the above formula.