Question

Mention the states of minimum and maximum potential energy situation of an electric dipole is an electrostatic field

Answer 1

Answer:

$\bigstar{\bold{U_{min}=-pE,Angle=0}}$

$\bigstar{\bold{U_{max}=pE,Angle=\pi}}$

Explanation:

$\Large{\underline{\underline{\bf{Given:}}}}$

• An electric dipole having dipole moment p is kept in a uniform electric field.

$\Large{\underline{\underline{\bf{To\:Find:}}}}$

• Minimum potential energy
• Maximum potential energy

$\Large{\underline{\underline{\bf{Solution:}}}}$

→ Consider an electric dipole having dipole moment p = q × 2a which is kept in a uniform electric field.

→ Here the torque experienced is given by

  τ = p E sin θ

→ Due to this torque, dipole moment produces a small displacement dθ where θ is the initial angle and Φ is the final angle.

→ Hence a work is done by the torque which is given by,

  dW = τ dθ

  dW = pE sin θ dθ

→ Hence the total work done is given by,

   $\sf{Total\:work\:done=\int\limits^\phi_\theta \,p\:E\:sin\theta d\theta}$

→ Taking the constants outside,

  $\sf{Total\:work\:done=p\:E\int\limits^\phi_\theta \,sin\theta d\theta}$

→ $\sf{We\:know\:that\int\limits {sin\:\theta} \, dx\theta=-cos\:\theta}$

→ Hence,

  $\sf{W=p\:E[-cos\:\theta]_\theta ^{\phi} }$

  W = -p E (cos θ - cos Φ)

→ Here the work done is stored as potential energy U

   U = -p E (cos θ - cos Φ)

→ Now let us consider some cases.

→ Case 1 :

  Initially dipole is in the same direction of magnnetic field.

  θ = 0°, Φ = Φ

  U = -pE (cos Φ - cos 0)

  U = -p E (cos Φ - 1)

→ Case 2:

  Initially the electric field and dipole moment is perpendicular to each      other.

 θ = 90°, Ф = Ф

 U = -pE (cosΦ - cos 90)

 U = -pE cos Φ

→ Now if Φ = 0,

  U = -pE cos 0

  U = -pE

→ Here the P.E is minimum, that is it is in stable equilibrium.

→ Hence the P.E is minimum when the angle between the dipole moment and electric field is 0.

$\boxed{\bold{U_{min}=-pE,Angle=0}}$

→ Now if Ф = 180°

  U = -pE (cos 180)

  U = -pE (-1)

  U = pE

→ Here P.E is maximum that is, it is in unstable equilibrium.

→ Hence the P.E is maximum when the angle between dipole moment and electric field is 180 or π.

$\boxed{\bold{U_{max}=pE,Angle=\pi}}$

→ If Ф = 90°

  U = -pE (cos 90)

  U = 0