Two identical electric point dipoles have dip | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The potential energy of two dipoles $\overrightarrow{p}_{1}$ and $\overrightarrow{p}_{2}$ separated by distance $r$ along the $x$-axis is given by

Vedclass · Accepted Answer

$\frac{p}{a} \sqrt{\frac{1}{2 \pi \varepsilon_{0} ma}}$

Vedclass · Answer

(C) The potential energy of two dipoles $\overrightarrow{p}_{1}$ and $\overrightarrow{p}_{2}$ separated by distance $r$ along the $x$-axis is given by $U = \frac{1}{4\pi\varepsilon_{0}} \left[ \frac{\overrightarrow{p}_{1} \cdot \overrightarrow{p}_{2} - 3(\overrightarrow{p}_{1} \cdot \hat{r})(\overrightarrow{p}_{2} \cdot \hat{r})}{r^{3}} ight]$.Here,$\overrightarrow{p}_{1} = p\hat{i}$,$\overrightarrow{p}_{2} = -p\hat{i}$,and $\hat{r} = \hat{i}$.So,$U = \frac{1}{4\pi\varepsilon_{0}} \left[ \frac{(p\hat{i}) \cdot (-p\hat{i}) - 3(p\hat{i} \cdot \hat{i})(-p\hat{i} \cdot \hat{i})}{a^{3}} ight] = \frac{1}{4\pi\varepsilon_{0}} \left[ \frac{-p^{2} - 3(p)(-p)}{a^{3}} ight] = \frac{1}{4\pi\varepsilon_{0}} \left[ \frac{2p^{2}}{a^{3}} ight] = \frac{p^{2}}{2\pi\varepsilon_{0}a^{3}}$.By conservation of energy,$KE_{i} + PE_{i} = KE_{f} + PE_{f}$.Initially,$KE_{i} = 0$ and $PE_{i} = \frac{p^{2}}{2\pi\varepsilon_{0}a^{3}}$.Finally,at infinite separation,$PE_{f} = 0$ and $KE_{f} = 2 	imes (\frac{1}{2}mv^{2}) = mv^{2}$.Thus,$0 + \frac{p^{2}}{2\pi\varepsilon_{0}a^{3}} = mv^{2} + 0$.$v^{2} = \frac{p^{2}}{2\pi\varepsilon_{0}ma^{3}}$.$v = \frac{p}{a} \sqrt{\frac{1}{2\pi\varepsilon_{0}ma}}$.

Similar Questions

At any point on the perpendicular bisector of the line joining two equal and opposite charges,which of the following is true?

$A$ wheel having mass $m$ has charges $+q$ and $-q$ on diametrically opposite points. It remains in equilibrium on a rough inclined plane in the presence of a vertical electric field $E$. Then the value of $E$ is

An electric dipole with dipole moment $p$ is placed with its axis at $30^{\circ}$ to a uniform electric field. The work done in rotating the dipole to a position where its axis is perpendicular to the field is

On the axis and on the equator of an electric dipole for all points . . . . . . .

If $E_{a}$ and $E_{q}$ represent the electric field intensity due to a short dipole at a point on its axial line and on the equatorial line at the same distance $r$ from the centre of the dipole,then

Vedclass Test Series

Exam Paper Generator

Online Exam Module