Question

An electric dipole moment of dipole is $\overrightarrow{p}=(-\hat{i}-3\hat{j}+2\hat{k})\times {10}^{-29}C.m$ is at the origin $(0,0,0)$. The electric field due to this dipole at $\hat{r}=(\hat{i}+3\hat{j}+5\hat{k})$ (note that $\overrightarrow{r}\cdot \overrightarrow{p}=0$) parallel is to:
A. $(-\hat{i}+3\hat{j}-2\hat{k})$
B. $(-\hat{i}-3\hat{j}+3\hat{k})$
C. $(\hat{i}+3\hat{j}-2\hat{k})$
D. $(\hat{i}-3\hat{j}-2\hat{k})$

Answer 1

Hint: The electric dipole moment is the product of the magnitude of the charge and the distance between the two charges. This is a useful concept for the study of dielectrics and other solid applications. The electric field of a dipole is always in the opposite direction (antiparallel) to that of the dipole moment.

As per the given data,
The electric dipole moment at origin is $\overrightarrow{p}=(-\hat{i}-3\hat{j}+2\hat{k})\times {10}^{-29}C.m$
The position vector is given as $\hat{r}=(\hat{i}+3\hat{j}+5\hat{k})$
The dot product of electric dipole moment and position vector is zero ($\overrightarrow{r}\cdot \overrightarrow{p}=0$)

Complete answer:
When two charges are kept parted by a certain distance it is known as a dipole. The product of the magnitude of the charge and the distance between the two charges is termed as an electric dipole moment.
Mathematically,
$p=qd$
Where,
$q$ is the magnitude of the charge
$d$ is the distance between two charges
The dot product is used to find the magnitude of the resultant quantity. Mathematically it is calculated as,
$$a\cdot b=\left|a\right|\left|b\right|\cos \theta$$
As it is mentioned in the question that the dot product of the electric dipole moment and the position vector is zero ($\overrightarrow{r}\cdot \overrightarrow{p}=0$).
So this can be written as,
$\begin{array}{rl}& \overrightarrow{r}\cdot \overrightarrow{p}=\left|r\right|\left|p\right|\cos \theta =0\\ & \Rightarrow \cos \theta =0\\ & \Rightarrow \theta ={\displaystyle \frac{\pi }{2}}\end{array}$
So here we can say that the electric field and position vector are perpendicular to each other.
The electric field of a dipole is given as,
$\overrightarrow{E}=-\lambda p$
Here the minus sign represents that the electric field is in the opposite (anti-parallel) direction to that of the electric dipole moment.
For the situation mention in the question the electric field of the dipole can be given as,
$\begin{array}{rl}& \overrightarrow{E}=-\overrightarrow{p}\\ & \Rightarrow \overrightarrow{E}=-(-\hat{i}-3\hat{j}+2\hat{k})\\ & \therefore \overrightarrow{E}=(\hat{i}+\hat{j}-2\hat{k})\end{array}$

Thus, from the above discussion, the correct option which satisfies the given question is Option C.

Note:
The concept of electric dipole moment is also useful in atoms and molecules where the effects of charge separation can be measured easily. We can also find the potential of a dipole at a distance point using the superposing the point charge potentials. The potential at the point decreases with an increase in the distance between the point and the dipole.