Question

A dipole is said to be in stable equilibrium when the angle between the electric field and dipole moment is

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Solution

The correct option is **A**

zero

**Step 1: Dipole moment**

- The dipole's moment is directed between the -ve and +ve charges. In equilibrium, the system's net force and torque should be zero.
- As a result, if potential energy is high, the equilibrium will be unstable, whereas if potential energy is low, the equilibrium will be stable.
- Torque is the amount of force that causes an object to revolve around an axis. Torque is a collection of vectors whose path is determined by the force acting on the axis.

**Step 2: Formula used**

The potential energy of the dipole will be equal to the $U=P{E}_{0}\mathrm{cos}\theta $

And for torque, the formula will be $\tau =P{E}_{0}\mathrm{sin}\theta $

$U$, will be the potential energy of the dipole

$\tau $, will be the torque

$P$, will be the dipole moment

${E}_{0}$, will be the uniform electric field

**Step 3: Calculating potential energy**

Let us assume that the dipole we are using is kept at two positions namely (a) and (b) which are placed in a constant electric field ${E}_{0}$ as is shown in the below diagram.

**In case (a): **When the angle between the dipole and electric field will be zero

Then, the potential energy of the dipole will be equal to the $U=-P{E}_{0}\mathrm{cos}\theta $

$U=-P{E}_{0}\mathrm{cos}{0}^{0}\phantom{\rule{0ex}{0ex}}U=-P{E}_{0}[\mathrm{cos}{0}^{0}=1]$

Therefore, we can say that the dipole will be in stable equilibrium.

And for torque, the formula will be $\tau =P{E}_{0}\mathrm{sin}\theta $

$\tau =P{E}_{0}\mathrm{sin}{0}^{0}\phantom{\rule{0ex}{0ex}}\tau =0[\mathrm{sin}{0}^{0}=0]$

**In case (b):** When the angle between the dipole and electric field will be ${180}^{o}$

Then, the potential energy of the dipole will be equal to the

$U=-P{E}_{0}\mathrm{cos}{180}^{0}[\mathrm{cos}{180}^{0}=-1]\phantom{\rule{0ex}{0ex}}U=P{E}_{0}$

Therefore, we can say that the dipole will be in an unstable equilibrium.

Now we will find the torque,

$\tau =P{E}_{0}\mathrm{sin}{180}^{0}\phantom{\rule{0ex}{0ex}}\tau =0[\mathrm{sin}{180}^{0}=0]$

Therefore, a dipole is said to be in stable equilibrium when the angle between the electric field and dipole moment is zero

**Hence, option A is the correct answer.**

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