First slide

Electric dipole

Question

A point particle of mass m is attached to one end of massless rigid non-conducting rod of length l. Another point particle of same mass is attached to the other end of the rod. The two particles carry equal charges +q and -q respectively. This arrangement is held in a region with the field E such that the rod makes an angle $θ$ with the field direction :

Difficult

A

The tension in rod remains constant
B

If q is very small, the rod executes simple harmonic motion of period $2 π \sqrt{\frac{ml}{2 qE}}$
C

At every instant, net force on the system is zero
D

Both (b) and ( c ) are correct

Solution

$τ = - pE sinθ$
For small displacement $sinθ \approx θ$
$Iα = - pEθ α \propto θ$
This shows that electric dipole shows angular S.H.M
In linear simple harmonic motion.
$ω = \sqrt{\frac{k}{m}} \begin{array}{l} \to spring constant \\ \to mass of oscillator \end{array}$
In angular S.H.M., $ω = \sqrt{\frac{pE}{I}} \Rightarrow \frac{2 π}{T} = \sqrt{\frac{pE}{I}}$
$∴$ I = M.I. of rod about the centre = $\frac{{ml}^{2}}{2} (∵ p = 2 ql)$
$\Rightarrow T = 2 π \sqrt{\frac{I}{pE}} ∴ T = 2 π \sqrt{\frac{ml}{2 qE}} Tension in the rod depends on the orientation of the dipole$

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