Question 161: Electrostatics & Capacitance

Question

Three charges $+2 q,+3 q$ and $-4 q$ are situated at $(0,-3 a),(2 a, 0)$ and $(-2 a, 0)$ respectively in the $x y$ plane. The resultant dipole moment about origin is $\_\_\_\_$ .

Accepted Answer

For a system of discrete point charges $q_i$ located at positions $\vec{r}_i$, the electric dipole moment $\vec{P}$ with respect to the origin is defined as the vector sum of the product of each charge and its position vector.
$$\overrightarrow{\mathrm{P}}=\sum_{\mathrm{i}} \mathrm{q}_{\mathrm{i}} \overrightarrow{\mathrm{r}}_{\mathrm{i}} $$
$$\Rightarrow $$ $$\overrightarrow{\mathrm{P}}=\mathrm{q}_1 \overrightarrow{\mathrm{r}}_1+\mathrm{q}_2 \overrightarrow{\mathrm{r}}_2+\mathrm{q}_3 \overrightarrow{\mathrm{r}}_3+\cdots$$
For charge $1\left(\mathrm{q}_1\right)$ :
$\mathrm{q}_1=+2 \mathrm{q}$ and it is at coordinates $(0,-3 \mathrm{a})$
So, position vector with respect to origin is, $=\overrightarrow{\mathrm{r}}_1=0 \hat{1}-3 a \hat{\jmath}=-3 a \hat{\jmath}$
So, electric dipole moment is, $\vec{p}_1=2 q(-3 a \hat{\jmath})=-6 a q \hat{\jmath}$
For charge $2\left(\mathrm{q}_2\right)$ :
$\mathrm{q}_2=+3 \mathrm{q}$ and it is at coordinates $(2 \mathrm{a}, 0)$
So, position vector with respect to origin is, $=\overrightarrow{\mathrm{r}}_2=2 a \hat{\mathrm{i}}-0 \hat{\mathrm{j}}=2 a \hat{\mathrm{i}}$
So, electric dipole moment is, $\vec{p}_2=3 q(2 a \hat{\mathrm{i}})=6 a q \hat{\mathrm{i}}$
For charge $3\left(\mathrm{q}_3\right)$ :
$\mathrm{q}_2=-4 \mathrm{q}$ and it is at coordinates $(-2 \mathrm{a}, 0)$
So, position vector with respect to origin is, $=\overrightarrow{\mathrm{r}}_3=-2 \mathrm{a} \hat{\mathrm{i}}-0 \hat{\mathrm{j}}=-2 \mathrm{a} \hat{\mathrm{i}}$
So, electric dipole moment is, $\vec{p}_3=-4 q(-2 a \hat{i})=8 a q \hat{i}$
So, the net electric dipole moment is the sum of the individual moments :
$$ \vec{P}=\vec{P}_1+\vec{P}_2+\vec{P}_3 $$
$$\Rightarrow $$ $\overrightarrow{\mathrm{P}}=(-6 q a \hat{\jmath})+(6 q a\hat{\imath})+(8 q a \hat{\imath})$
$$\Rightarrow $$ $\overrightarrow{\mathrm{P}}=14 q a \hat{\imath}-6 q a \hat{\jmath}$
$$\Rightarrow $$ $$ \overrightarrow{\mathrm{P}}=2 \mathrm{qa}(7 \hat{\imath}-3 \hat{\jmath}) $$
Hence, the correct option is (A).

Answer

$2 q a(7 \hat{i}-3 \hat{j})$

Answer

$2 q a(3 \hat{j}-7 \hat{i})$

Answer

$2 q a(3 \hat{i}-7 \hat{j})$

Answer

$2 q a(3 \hat{j}-\hat{i})$

Electrostatics & Capacitance

Three charges +2 q,+3 q and -4 q are situated at (0,-3 a),(2 a, 0) and (-2 a, 0) respectively in the x y plane. The resultant dipole moment about origin is \_\_\_\_ .

Choose the correct answer: