Electric Dipole And Torque

A dipole in a uniform field feels no net force but experiences a torque given by $\tau =pE\sin \theta$, where $p$ is the dipole moment, $E$ the field, and $\theta$ the angle between them, as derived in NCERT Class 12, Chapter 1 (Electric Charges and Fields). Substituting $p=4\times 10^{-9}$ C·m, $E=5\times 10^4$ N/C, and $\theta =30^{\circ }$ with $\sin 30^{\circ }=0.5$ gives $\tau =(4\times 10^{-9})(5\times 10^4)(0.5)$. First $4\times 5=20$ and $10^{-9}\times 10^4=10^{-5}$, producing $20\times 10^{-5}=2\times 10^{-4}$; multiplying by $0.5$ yields $1\times 10^{-4}$ N·m. The value $2\times 10^{-4}$ N·m is wrong because it omits the $\sin 30^{\circ }$ factor. The value $1.73\times 10^{-4}$ N·m is wrong because it uses $\sin 60^{\circ }$ instead of $\sin 30^{\circ }$. The value 0 N·m is wrong because torque vanishes only when the dipole is aligned at $0^{\circ }$ or $180^{\circ }$, not at $30^{\circ }$. Physically the torque tends to rotate the dipole until it aligns with the field, and it is greatest when the dipole is perpendicular to the field and vanishes when parallel. A magnitude and unit check confirms the product of C·m and N/C gives N·m, and the small dipole moment yields a tiny torque consistent with the order of $10^{-4}$ obtained here.