Derive an expression for the torque acting on an electric dipole placed in a uniform electric field. Under what conditions is this torque maximum and minimum?

Q: Derive an expression for the torque acting on an electric dipole placed in a uniform electric field. Under what conditions is this torque maximum and minimum?

Derivation of Torque on an Electric Dipole: \nConsider an electric dipole consisting of two charges $+q$ and $-q$ separated by a distance $2a$, placed in a uniform electric field $\vec{E}$ making an angle $\theta$ with the direction of the electric field. Forces on charges: Force on charge $+q$ is $\vec{F}1 = +q\vec{E}$ (in direction of $\vec{E}$). Force on charge $-q$ is $\vec{F}2 = -q\vec{E}$ (opposite to direction of $\vec{E}$). Net Force: The net translational force is $\vec{F}{net} = q\vec{E} - q\vec{E} = 0$. However, these equal and opposite forces act at different lines of action, forming a couple that exerts a torque. Torque Calculation: $$\tau = \text{Force} \times \text{Perpendicular distance between forces}$$ From the geometry, perpendicular distance $= 2a \sin\theta$. $$\tau = (qE) \times (2a \sin\theta) = (q \cdot 2a) E \sin\theta$$ Since dipole moment $p = q \cdot 2a$, we get: $$\tau = pE \sin\theta \quad \text{or} \quad \vec{\tau} = \vec{p} \times \vec{E}$$ Special Cases: Maximum Torque: Torque is maximum when $\theta = 90^\circ$ (dipole is perpendicular to the field). Then $\tau{max} = pE$. Minimum Torque: Torque is zero (minimum) when $\theta = 0^\circ$ (stable equilibrium) or $\theta = 180^\circ$ (unstable equilibrium). Then $\tau{min} = 0$.

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Derivation of Torque on an Electric Dipole: 
Consider an electric dipole consisting of two charges $+q$ and $-q$ separated by a distance $2a$, placed in a uniform electric field $\vec{E}$ making an angle $	heta$ with the direction of the electric field. Forces on charges: Force on charge $+q$ is $\vec{F}1 = +q\vec{E}$ (in direction of $\vec{E}$). Force on charge $-q$ is $\vec{F}2 = -q\vec{E}$ (opposite to direction of $\vec{E}$). Net Force: The net translational force is $\vec{F}{net} = q\vec{E} - q\vec{E} = 0$. However, these equal and opposite forces act at different lines of action, forming a couple that exerts a torque. Torque Calculation: $$	au = 	ext{Force} 	imes 	ext{Perpendicular distance between forces}$$ From the geometry, perpendicular distance $= 2a \sin	heta$. $$	au = (qE) 	imes (2a \sin	heta) = (q \cdot 2a) E \sin	heta$$ Since dipole moment $p = q \cdot 2a$, we get: $$	au = pE \sin	heta \quad 	ext{or} \quad \vec{	au} = \vec{p} 	imes \vec{E}$$ Special Cases: Maximum Torque: Torque is maximum when $	heta = 90^\circ$ (dipole is perpendicular to the field). Then $	au{max} = pE$. Minimum Torque: Torque is zero (minimum) when $	heta = 0^\circ$ (stable equilibrium) or $	heta = 180^\circ$ (unstable equilibrium). Then $	au{min} = 0$.

CBSE · Class 12 · Physics · Electric Charges and FieldsDerive an expression for the torque acting on an electric dipole placed in a uniform electric field. Under what conditions is this torque maximum and minimum?