Question
Three point charges $$+ q, - 2q$$ and $$+q$$ are placed at points $$\left( {x = 0,y = a,z = 0} \right),\left( {x = 0,y = 0,z = 0} \right)$$ and $$\left( {x = a,y = 0,z = 0} \right),$$ respectively. The magnitude and direction of the electric dipole moment vector of this charge assembly are
A.
$$\sqrt 2 qa$$ along $$+y$$ direction
B.
$$\sqrt 2 aq$$ along the line joining points $$\left( {x = 0,y = 0,z = 0} \right)$$ and $$\left( {x = a,y = a,z = 0} \right)$$
C.
$$qa$$ along the line joining points $$\left( {x = 0,y = 0,z = 0} \right)$$ and $$\left( {x = a,y = a,z = 0} \right)$$
D.
$$\sqrt 2 aq$$ along $$+ x$$ direction
Answer :
$$\sqrt 2 aq$$ along the line joining points $$\left( {x = 0,y = 0,z = 0} \right)$$ and $$\left( {x = a,y = a,z = 0} \right)$$
Solution :
Choose the three coordinate axes as $$X,Y$$ and $$Z$$ and plot the charges with the given coordinates as shown. $$O$$ is the origin at which $$- 2q$$ charge is placed. The system is equivalent to two dipoles along $$x$$ and $$y$$-directions respectively. The dipole moments of two dipoles are shown in figure.
The resultant dipole moment will be directed along $$OP$$ where coordinate of $${P_{{\text{net}}}}$$ is given by $$\left( {a,a,0} \right).$$ The magnitude of resultant dipole moment is $${P_{{\text{net}}}} = \sqrt {{p^2} + {p^2}} $$
$$\eqalign{ & = \sqrt {{{\left( {qa} \right)}^2} + {{\left( {qa} \right)}^2}} \cr & = \sqrt 2 qa \cr} $$
Choose the three coordinate axes as $$X,Y$$ and $$Z$$ and plot the charges with the given coordinates as shown. $$O$$ is the origin at which $$- 2q$$ charge is placed. The system is equivalent to two dipoles along $$x$$ and $$y$$-directions respectively. The dipole moments of two dipoles are shown in figure.
The resultant dipole moment will be directed along $$OP$$ where coordinate of $${P_{{\text{net}}}}$$ is given by $$\left( {a,a,0} \right).$$ The magnitude of resultant dipole moment is $${P_{{\text{net}}}} = \sqrt {{p^2} + {p^2}} $$
$$\eqalign{ & = \sqrt {{{\left( {qa} \right)}^2} + {{\left( {qa} \right)}^2}} \cr & = \sqrt 2 qa \cr} $$
