Electric Dipole And Torque

Question

An electric dipole of moment 4 	imes 10^{-9} C·m is placed at 30 degrees to a uniform field of 5 	imes 10^4 N/C, so what torque acts on it?

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A dipole in a uniform field feels no net force but experiences a torque given by $	au = pE\sin	heta$, where $p$ is the dipole moment, $E$ the field, and $	heta$ the angle between them, as derived in NCERT Class 12, Chapter 1 (Electric Charges and Fields). Substituting $p = 4	imes10^{-9}$ C·m, $E = 5	imes10^{4}$ N/C, and $	heta = 30^\circ$ with $\sin30^\circ = 0.5$ gives $	au = (4	imes10^{-9})(5	imes10^{4})(0.5)$. First $4	imes5 = 20$ and $10^{-9}	imes10^{4} = 10^{-5}$, producing $20	imes10^{-5} = 2	imes10^{-4}$; multiplying by $0.5$ yields $1	imes10^{-4}$ N·m. The value $2	imes10^{-4}$ N·m is wrong because it omits the $\sin30^\circ$ factor. The value $1.73	imes10^{-4}$ N·m is wrong because it uses $\sin60^\circ$ instead of $\sin30^\circ$. 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.

Electric Dipole And Torque

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