Question

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

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 $\vec{p}=\left( -\hat{i}-3\hat{j}+2\hat{k} \right)\times {{10}^{-29}}C.m$
The position vector is given as $\hat{r}=\left( \hat{i}+3\hat{j}+5\hat{k} \right)$
The dot product of electric dipole moment and position vector is zero ($\vec{r}\cdot \vec{p}=0$)

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 ($\vec{r}\cdot \vec{p}=0$).
So this can be written as,
$\begin{align}
  & \vec{r}\cdot \vec{p}=\left| r \right|\left| p \right|\cos \theta =0 \\
 & \Rightarrow \cos \theta =0 \\
 & \Rightarrow \theta =\dfrac{\pi }{2} \\
\end{align}$
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,
$\vec{E}=-\lambda p$
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,
$\begin{align}
  & \vec{E}=-\vec{p} \\
 & \Rightarrow \vec{E}=-\left( -\hat{i}-3\hat{j}+2\hat{k} \right) \\
 & \therefore \vec{E}=\left( \hat{i}+\hat{j}-2\hat{k} \right) \\
\end{align}$

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.