Consider three charges $q_{1}, q_{2}, q_{3}$ each equal to $q$ at the vertices of an equilateral triangle of side $l .$ What is the force on a charge $Q$ (with the same sign as $q$ ) placed at the centroid of the triangle, as shown in Figure
Consider three charges $q_{1}, q_{2}, q_{3}$ each equal to $q$ at the vertices of an equilateral triangle of side $l .$ What is the force on a charge $Q$ (with the same sign as $q$ ) placed at the centroid of the triangle, as shown in Figure
In the given equilateral triangle $ABC$ of sides of length $l$, if Iraw a perpendicular $AD$ to the side $BC,$
$A D=A C \cos 30^{\circ}=(\sqrt{3} / 2) l$ and the distance $AO$ of the centroid $O$ from $A$ is $(2 / 3) AD =(1 / \sqrt{3})$ $l$. By symmatry $AO = BO = CO$
Thus,
Force $F _{1}$ on $Q$ due to charge $q$ at $A =\frac{3}{4 \pi \varepsilon_{0}} \frac{ Q q}{l^{2}}$ along $AO$
Force $F _{2}$ on $Q$ due to charge $q$ at $B =\frac{3}{4 \pi \varepsilon_{0}} \frac{ Q q}{l^{2}}$ along $BO$
Force $F_{3}$ on $Q$ due to charge $q$ at $C=\frac{3}{4 \pi \varepsilon_{0}} \frac{Q q}{l^{2}}$ along $CO$
The resultant of forces $F _{2}$ and $F _{3}$ is $\frac{3}{4 \pi \varepsilon_{0}} \frac{Q q}{l^{2}}$ along $OA$. by the parallelogram law. Therefore, the total force on $g=\frac{3}{4 \pi \varepsilon_{0}} \frac{Q q}{l^{2}}(\hat{ r }-\hat{ r })$
$=0,$ where $\hat{ r }$ is the unit vector along $OA$.
It is clear also by symmetry that the three forces will sum to zero. Suppose that the resultant force was non-zero but in some direction. Consider what would happen if the system was rotated through $60^{\circ}$ about $O$.
Similar Questions
Two free positive charges $4q$ and $q$ are a distance $l$ apart. What charge $Q$ is needed to achieve equilibrium for the entire system and where should it be placed form charge $q$ ?
Two free positive charges $4q$ and $q$ are a distance $l$ apart. What charge $Q$ is needed to achieve equilibrium for the entire system and where should it be placed form charge $q$ ?
Two fixed charges $4\,Q$ (positive) and $Q$ (negative) are located at $A$ and $B$, the distance $AB$ being $3$ $m$.
Two fixed charges $4\,Q$ (positive) and $Q$ (negative) are located at $A$ and $B$, the distance $AB$ being $3$ $m$.
The electric field between the two spheres of a charged spherical condenser
The electric field between the two spheres of a charged spherical condenser
Consider a system of three charges $\frac{q}{3},\frac{q}{3}$ and $\frac{-2q}{3}$ placed at points $A,B$ and $C$, respectively, as shown in the figure. Take $O$ to be the centre of the circle of radius $R$ and angle $CAB = 60^o$
Consider a system of three charges $\frac{q}{3},\frac{q}{3}$ and $\frac{-2q}{3}$ placed at points $A,B$ and $C$, respectively, as shown in the figure. Take $O$ to be the centre of the circle of radius $R$ and angle $CAB = 60^o$
Four charge $Q _1, Q _2, Q _3$, and $Q _4$, of same magnitude are fixed along the $x$ axis at $x =-2 a - a ,+ a$ and $+2 a$, respectively. A positive charge $q$ is placed on the positive $y$ axis at a distance $b > 0$. Four options of the signs of these charges are given in List-$I$ . The direction of the forces on the charge q is given in List-$II$ Match List-$1$ with List-$II$ and select the correct answer using the code given below the lists.$Image$
List-$I$
List-$II$
$P.$ $\quad Q _1, Q _2, Q _3, Q _4$, all positive
$1.\quad$ $+ x$
$Q.$ $\quad Q_1, Q_2$ positive $Q_3, Q_4$ negative
$2.\quad$ $-x$
$R.$ $\quad Q_1, Q_4$ positive $Q_2, Q_3$ negative
$3.\quad$ $+ y$
$S.$ $\quad Q_1, Q_3$ positive $Q_2, Q_4$ negative
$4.\quad$ $-y$
Four charge $Q _1, Q _2, Q _3$, and $Q _4$, of same magnitude are fixed along the $x$ axis at $x =-2 a - a ,+ a$ and $+2 a$, respectively. A positive charge $q$ is placed on the positive $y$ axis at a distance $b > 0$. Four options of the signs of these charges are given in List-$I$ . The direction of the forces on the charge q is given in List-$II$ Match List-$1$ with List-$II$ and select the correct answer using the code given below the lists.$Image$
| List-$I$ | List-$II$ |
| $P.$ $\quad Q _1, Q _2, Q _3, Q _4$, all positive | $1.\quad$ $+ x$ |
| $Q.$ $\quad Q_1, Q_2$ positive $Q_3, Q_4$ negative | $2.\quad$ $-x$ |
| $R.$ $\quad Q_1, Q_4$ positive $Q_2, Q_3$ negative | $3.\quad$ $+ y$ |
| $S.$ $\quad Q_1, Q_3$ positive $Q_2, Q_4$ negative | $4.\quad$ $-y$ |
- [IIT 2014]