Question

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

A

zero

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B

${180}^{o}$

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C

${45}^{o}$

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D

${90}^{o}$

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Solution

## The correct option is A zeroStep 1: Dipole momentThe 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 usedThe 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$$U$, will be the potential energy of the dipole$\tau$, will be the torque$P$, will be the dipole moment${E}_{0}$, will be the uniform electric fieldStep 3: Calculating potential energyLet us assume that the dipole we are using is kept at two positions namely (a) and (b) which are placed in a constant electric field ${E}_{0}$ as is shown in the below diagram.In case (a): When the angle between the dipole and electric field will be zeroThen, the potential energy of the dipole will be equal to the $U=-P{E}_{0}\mathrm{cos}\theta$$U=-P{E}_{0}\mathrm{cos}{0}^{0}\phantom{\rule{0ex}{0ex}}U=-P{E}_{0}\left[\mathrm{cos}{0}^{0}=1\right]$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\left[\mathrm{sin}{0}^{0}=0\right]$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$U=-P{E}_{0}\mathrm{cos}{180}^{0}\left[\mathrm{cos}{180}^{0}=-1\right]\phantom{\rule{0ex}{0ex}}U=P{E}_{0}$Therefore, we can say that the dipole will be in an unstable equilibrium.Now we will find the torque,$\tau =P{E}_{0}\mathrm{sin}{180}^{0}\phantom{\rule{0ex}{0ex}}\tau =0\left[\mathrm{sin}{180}^{0}=0\right]$Therefore, a dipole is said to be in stable equilibrium when the angle between the electric field and dipole moment is zeroHence, option A is the correct answer.

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