Question 2: (a) Define an Electric Dipole and Electric Dipole Moment. Is dipole moment a scalar or vector quantity? State its SI unit. (b) Derive an expression for the electric field intensity at a point situated on the axial position (end-on position) of an electric dipole. (c) What happens when an electric dipole is placed in a uniform external electric field? Write the expression for the torque acting on it.

Question

Accepted Answer

(a) Definitions and Properties Electric Dipole: A system consisting of two equal and opposite point charges ($+q$ and $-q$) separated by a very small finite distance ($2l$) is called an electric dipole. Electric Dipole Moment ($\vec{p}$): It is defined as the product of the magnitude of either charge ($q$) and the distance ($2l$) between them. $$\vec{p} = q 	imes 2\vec{l}$$ Nature: It is a vector quantity. Its direction is conventionally from the negative charge ($-q$) to the positive charge ($+q$) along the dipole axis. SI Unit: Coulomb-metre ($	ext{C}\cdot	ext{m}$). --- (b) Expression for Electric Field on Axial Line (End-on Position) Setup and Geometry: Let an electric dipole $AB$ consist of charges $-q$ at point $A$ and $+q$ at point $B$, separated by distance $2l$. Let $O$ be the center of the dipole. So, $OA = OB = l$. Consider a point $P$ on the axial line at a distance $r$ from the center $O$. Distance of point $P$ from charge $+q$ (at $B$): $BP = r - l$. Distance of point $P$ from charge $-q$ (at $A$): $AP = r + l$. Electric Field Components at Point $P$: Electric field intensity at $P$ due to charge $+q$ (directed away from dipole along $\vec{BP}$): $$E1 = \frac{1}{4\pi\varepsilon0} \frac{q}{(r - l)^2}$$ Electric field intensity at $P$ due to charge $-q$ (directed towards dipole along $\vec{PA}$): $$E2 = \frac{1}{4\pi\varepsilon0} \frac{q}{(r + l)^2}$$ Net Electric Field ($E{	ext{axial}}$): Since $E1 > E2$ (because point $P$ is closer to $+q$), the resultant electric field $E$ acts along the direction of $E1$ (away from dipole along the axis): $$E{	ext{axial}} = E1 - E2$$ $$E{	ext{axial}} = \frac{q}{4\pi\varepsilon0} \left \frac{1}{(r - l)^2} - \frac{1}{(r + l)^2} ight$$ $$E{	ext{axial}} = \frac{q}{4\pi\varepsilon0} \left \frac{(r + l)^2 - (r - l)^2}{(r^2 - l^2)^2} ight$$ $$E{	ext{axial}} = \frac{q}{4\pi\varepsilon0} \left \frac{4rl}{(r^2 - l^2)^2} ight$$ $$E{	ext{axial}} = \frac{1}{4\pi\varepsilon0} \left \frac{2 \cdot (q \cdot 2l) \cdot r}{(r^2 - l^2)^2} ight$$ Substituting dipole moment $p = q \cdot 2l$: $$E{	ext{axial}} = \frac{1}{4\pi\varepsilon0} \frac{2pr}{(r^2 - l^2)^2}$$ Special Case (Short Dipole, $r \gg l$): If the dipole is very short or the point $P$ is far away such that $r \gg l$, then $l^2$ can be neglected in comparison to $r^2$: $$E{	ext{axial}} \approx \frac{1}{4\pi\varepsilon0} \frac{2pr}{r^4} = \frac{1}{4\pi\varepsilon0} \frac{2p}{r^3}$$ Direction: The net electric field vector $\vec{E}{	ext{axial}}$ is along the direction of dipole moment $\vec{p}$. --- (c) Behavior in Uniform External Electric Field When an electric dipole is placed in a uniform external electric field $\vec{E}$, equal and opposite forces ($F = qE$) act on its two charges. Net Force: $F{	ext{net}} = 0$ (hence no translational motion). Torque: The two forces form a couple that tries to align the dipole parallel to the electric field direction. Expression for Torque: $$	au = p E \sin	heta$$ In vector form: $\vec{	au} = \vec{p} 	imes \vec{E}$ (where $	heta$ is the angle between $\vec{p}$ and $\vec{E}$).