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Question: An electric dipole of length 2 cm, when placed with its axis making an angle of \(60^\circ \) with u...

An electric dipole of length 2 cm, when placed with its axis making an angle of 60∘60^\circ with uniform electric field, experiences a torque of 83Nm8\sqrt 3 Nm. Calculate the potential energy of the dipole, if it has a charge of ±4nC \pm 4nC:

Explanation

Solution

Dipole is the separation between the two opposite charges .The electric dipole is the pair of equal and opposite charges +q + q and −q - q separated by a distance 2a. The dipole moment is defined as the total polarity of the system.

Formula used:
The formula of dipole moment is given by,
P=q×(2a)P = q \times \left( {2a} \right)
Where dipole moment is P charge is q and the distance between the two charges is 2a.
The formula of the torque on the dipole in uniform electric field is given by,
τ=P⋅Esin⁡θ\tau = P \cdot E\sin \theta
Where τ\tau is the torque the dipole moment is P the electric field is E and the angle between the dipole and the electric field isθ\theta .
The formula of the potential energy of the of the dipole is given by,
U=−P⋅Ecos⁡θU = - P \cdot E\cos \theta
Where potential energy is U the dipole moment is P the electric field is E and the angle between dipole and the electric field isθ\theta .

Complete step by step answer:
It is given in the problem that an electric dipole of length 2 cm, when placed with its axis making an angle of 60∘60^\circ with uniform electric field, experiences a torque of 83Nm8\sqrt 3 Nm. If it has a charge of ±4nC \pm 4nC.
Then we need to find the potential energy of the dipole.
First of all we need to calculate the dipole moment,
The formula of dipole moment is given by,
P=q×(2a)P = q \times \left( {2a} \right)
Where dipole moment is P charge is q and the distance between the two charges is 2a.
The charge is given by q=4×10−9Cq = 4 \times {10^{ - 9}}C and the separation is 2a=2×2×10−2m2a = 2 \times 2 \times {10^{ - 2}}m.
The dipole moment is equal to,
P=(4×10−9)⋅(2×2×10−2)P = \left( {4 \times {{10}^{ - 9}}} \right) \cdot \left( {2 \times 2 \times {{10}^{ - 2}}} \right)
P=16×10−11CmP = 16 \times {10^{ - 11}}Cm
The formula of the torque on the dipole in uniform electric field is given by,
τ=P⋅Esin⁡θ\tau = P \cdot E\sin \theta
Where τ\tau is the torque the dipole moment is P the electric field is E and the angle between the dipole and the electric field is θ\theta .
Electric field is equal to,
τ=P⋅Esin⁡θ\tau = P \cdot E\sin \theta
E=τP⋅sin⁡θE = \dfrac{\tau }{{P \cdot \sin \theta }}………eq. (1)
The formula of the potential energy of the of the dipole is given by,
U=−P⋅Ecos⁡θU = - P \cdot E\cos \theta
Where potential energy is U the dipole moment is P the electric field is E and the angle between dipole and the electric field isθ\theta .
Replace the value of electric field in the above equation,
⇒U=−P⋅Ecos⁡θ\Rightarrow U = - P \cdot E\cos \theta
⇒U=−P⋅cos⁡θ⋅(τP⋅sin⁡θ)\Rightarrow U = - P \cdot \cos \theta \cdot \left( {\dfrac{\tau }{{P \cdot \sin \theta }}} \right)
⇒U=−cos⁡θ⋅(τsin⁡θ)\Rightarrow U = - \cos \theta \cdot \left( {\dfrac{\tau }{{\sin \theta }}} \right)
⇒U=−(τtan⁡θ)\Rightarrow U = - \left( {\dfrac{\tau }{{\tan \theta }}} \right)
⇒U=−(83tan⁡60∘)\Rightarrow U = - \left( {\dfrac{{8\sqrt 3 }}{{\tan 60^\circ }}} \right)
⇒U=−8Nm\Rightarrow U = - 8Nm
The potential energy is equal to U=−8NmU = - 8Nm.

Note: The dipole is the separation between the two opposite charges whereas dipole moment is the total polarity in the system. The potential energy is the energy that the dipole acquires under the influence of the electric field.