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Question
An electric dipole has a fixed dipole moment `vec "p"`, which makes angle `theta` with respect to x-axis. When subjected to an electric field `vec"E"_1 = "E"hat"i"`, it experiences a torque `vec "T"_1 = tau hat"k"`. When subjected to another electric field `vec "E"_2 = sqrt3 "E"_1 hat "j"` it experiences torque `vec "T"_2 = ~ vec "T"_1` The angle `theta` is ____________.
Options
60°
90°
30°
45°
Solution
An electric dipole has a fixed dipole moment `vec "p"`, which makes angle `theta` with respect to x-axis. When subjected to an electric field `vec"E"_1 = "E"hat"i"`, it experiences a torque `vec "T"_1 = tau hat"k"`. When subjected to another electric field `vec "E"_2 = sqrt3 "E"_1 hat "j"` it experiences torque `vec "T"_2 = ~ vec "T"_1` The angle `theta` is 60°.
Explanation:
`vec tau = vec"p" xx vec "E"`
`therefore vec "T"_2 + vec "T"_1 ("p"_"x" hat"i" + "p"_"y" hat "j") xx ("E" hat "i" + sqrt "E" hat "j")`
`therefore tau hat "k" - tau hat "k" = "p"_"x" xx sqrt3 "E" hat "k" + "p"_"y" "E" (- hat "k")`
`therefore 0 = "E" hat "k" (sqrt3 "p"_"x" - "p"_"y")`
`therefore "p"_"x"/"p"_"y" = sqrt3`
`therefore "tan" theta = sqrt3`
`therefore theta = 60°`
