question_answer

A charge Q located at a point \[\vec{r}\] is in equilibrium under the combined electric field of three charges \[{{\mathbf{q}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{q}}_{\mathbf{2}}}\mathbf{,}{{\mathbf{q}}_{\mathbf{3}}}\mathbf{.}\]charges \[{{\mathbf{q}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{q}}_{\mathbf{2}}}\] are located at points \[{{\mathbf{\vec{r}}}_{\mathbf{1}}}\] and \[{{\mathbf{\vec{r}}}_{2}}\] respectively, find the direction of the force on Q, due to \[{{\mathbf{q}}_{\mathbf{3}}}\] in terms of\[{{\mathbf{q}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{q}}_{\mathbf{2}}}\mathbf{,}{{\mathbf{\vec{r}}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{\vec{r}}}_{\mathbf{2}}}\] and \[\mathbf{\vec{r}}\].

Answer:

                The forces exerted by \[{{q}_{1}}\] and \[{{q}_{2}}\] on Q are shown in Fig.                 According to Coulomb's law, force exerted by \[{{q}_{1}}\] on \[\text{Q}\] is \[{{\vec{F}}_{1}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{Q{{q}_{1}}}{|\vec{r}-{{{\vec{r}}}_{1}}{{|}^{2}}}\frac{\vec{r}-{{{\vec{r}}}_{1}}}{|\vec{r}-{{{\vec{r}}}_{1}}|}\] \[=\frac{1}{4\pi {{\varepsilon }_{0}}}Q{{q}_{1}}\frac{\vec{r}-{{{\vec{r}}}_{1}}}{|\vec{r}-{{{\vec{r}}}_{1}}{{|}^{3}}}\] Force exerted by \[{{q}_{2}}\]on \[Q\]is \[{{\vec{F}}_{2}}=\frac{1}{4\pi {{\varepsilon }_{0}}}Q{{q}_{2}}\frac{\vec{r}-{{{\vec{r}}}_{2}}}{|\vec{r}-{{{\vec{r}}}_{2}}{{|}^{3}}}\] According to the principle of superposition, total force exerted by \[{{q}_{1}}\] and \[{{q}_{2}}\] on \[Q\] is \[\vec{F}={{\vec{F}}_{1}}+{{\vec{F}}_{2}}\]                 \[=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ Q{{q}_{1}}\frac{\vec{r}-{{{\vec{r}}}_{1}}}{|\vec{r}-{{{\vec{r}}}_{1}}{{|}^{3}}}+Q{{q}_{2}}\frac{\vec{r}-{{{\vec{r}}}_{2}}}{|\vec{r}-{{{\vec{r}}}_{2}}{{|}^{3}}} \right]\] The charge Q will be in equilibrium if the force exerted by \[{{q}_{3}}\] is equal and opposite to the combined force exerted by \[{{q}_{1}}\] and\[{{q}_{2}}\]. \[\therefore \] Force exerted by \[{{q}_{3}}\] on \[Q=-\vec{F}\] \[=-\frac{Q}{4\pi {{\varepsilon }_{0}}}\left[ {{q}_{1}}\frac{\vec{r}-{{{\vec{r}}}_{1}}}{|\vec{r}-{{{\vec{r}}}_{1}}{{|}^{3}}}+{{q}_{2}}\frac{\vec{r}-{{{\vec{r}}}_{2}}}{|\vec{r}-{{{\vec{r}}}_{2}}{{|}^{3}}} \right]\] \[=\frac{Q}{4\pi {{\varepsilon }_{0}}}\left[ {{q}_{1}}\frac{{{{\vec{r}}}_{1}}-\vec{r}}{|\vec{r}-{{{\vec{r}}}_{1}}{{|}^{3}}}+{{q}_{2}}\frac{{{{\vec{r}}}_{2}}-\vec{r}}{|\vec{r}-{{{\vec{r}}}_{2}}{{|}^{3}}} \right]\] The above vector gives the direction of the force on \[Q\] due to \[{{q}_{3}}\].