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Question
A dipole with its charges, - q and + q located at the points (O, -b, O) and (O, +b, O) is present in a uniform electric field E. The equipotential surfaces of this field are planes parallel to the yz plane.
(a) What is the direction of the electric field E?
(b) How much torque would the dipole experience in this field?
Solution
(a) Given, the equipotentials of the external uniform electric field are planes parallel to the yz plane, the electric field `vec "E" = +- "E" hat"i"` that is, `vec "E"` is parallel to the x-axis.
A dipole in an external electric field along x-axis
(b) The above diagram, the dipole moment, `vec"p" = "q"(2"b") hat"j"`
The torque on this dipole,
`vec τ = vec"p" xx vec "E" = (2 "qb"hat"j") xx (+- "E"hat"i") = (2"qbE")(hat"j" xx hat "i")`
Since `hat"j" xx hat "i" = - bar"k"`
`vec τ = (+- 2"qbE")(-hat"k") = (2"qbE")(+- hat"k")`
So that the magnitude of the torque is τ = 2qbE.
If `vec"E"` is in the direction of the + x-axis, the torque `vec τ` is in the direction of - z-axis, while if `vec "E"` s in the direction of the - x-axis, the torque `vec tau` is in the direction of + z-axis.
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Answer in brief:
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