JEE Advanced 2020 · Paper 2 · Physics Q5

← JEE Advanced 2020 · Paper 2

PhysicsInteger+3 / -1Question 5 of 54

A point charge $q$ of mass $m$ is suspended vertically by a string of length $l$ . A point dipole of dipole moment $\vec p$ is now brought towards $q$ from infinity so that the charge moves away. The final equilibrium position of the system including the direction of the dipole, the angles and distances is shown in the figure below. If the work done in bringing the dipole to this position is $N\times(mgh)$ , where $g$ is the acceleration due to gravity, then the value of $N$ is ______. (Note that for three coplanar forces keeping a point mass in equilibrium, $\dfrac{F}{\sin\theta}$ is the same for all forces, where $F$ is any one of the forces and $\theta$ is the angle between the other two forces)

A string of length l hangs from a fixed point; the charge q at its end is deflected by angle α from the vertical and has risen by height h. The dipole p sits at the charge's original position (a distance l directly below the support), pointing along the line towards q, which is a distance 2l sin(α/2) away.

Your answer

Answer and worked solution

Official answer: 2

Geometry. The pivot, the dipole (at the charge's original position, a distance $l$ below the pivot) and the charge form an isosceles triangle. Its apex angle is $\alpha$ at the pivot, and its base $r=2l\sin\frac\alpha2$ joins the dipole to the charge.

Force on the charge. The dipole points at $q$ , so on its axis $E=\dfrac{2kp}{r^3}$ and the repulsion is $F=\dfrac{2kpq}{r^3}$ , directed along the base.

Equilibrium (Lami). Measured at the charge:

the angle between $T$ and $mg$ is $180^\circ-\alpha$ ;
the angle between $F$ and $mg$ is $90^\circ+\frac\alpha2$ .

$\frac F{\sin(180^\circ-\alpha)}=\frac{mg}{\sin(90^\circ+\frac\alpha2)}\implies F=\frac{mg\sin\alpha}{\cos\frac\alpha2}=2mg\sin\frac\alpha2.$

Interaction energy.
$U=qV_{dip}=\frac{kpq}{r^2}=\frac{Fr}2=2mg\sin\tfrac\alpha2\cdot l\sin\tfrac\alpha2=mgl(1-\cos\alpha)=mgh.$

Work done. The work in bringing the dipole equals the gain in gravitational plus electrostatic energy:
$W=mgh+U=2mgh\implies N=2.$
Answer: 2

Topics:Electric dipole Potential energy of charge systems Free body diagrams

← Previous question Next question →

JEE Advanced 2020 · Paper 2, Physics question 5