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Translating scalar torque quantities to their vector analogues (RE: Dipoles)

  1. Apr 8, 2012 #1
    My question is at the bottom of this post

    PREAMBLE:

    If a dipole is turned by an angle θ (in a uniform electric field) then the torque applied on the dipole by the electric field will be:

    τ = 2.q.a.E.sin(-θ) = -2.q.a.E.sin(θ)

    with the negative sign referring to it being a "restoring" torque. This negative sign is important in:

    -dU = ∫τ.dθ = -2.q.a.E.∫sin(θ).dθ

    IN TERMS OF VECTORS:

    In τ = p x E

    p = 2.q.a (in the direction of a), and

    E.sin(θ) is the "x E" part of τ = p x E

    MY QUESTION:
    Where is the negative sign gone in the vector equation? what accounts for it?
     
    Last edited: Apr 9, 2012
  2. jcsd
  3. Apr 9, 2012 #2
    I have this now. (I omitted a negative sign)

    If I rotate the dipole by θ from equilibrium then I've applied a torque:

    τ = p x E, or
    τ = 2qa.E.sin(θ)

    The restoring torque due to the (uniform) electric field will be to reduce θ (and thus restore equilibrium)

    τ = 2qa.E.sin(-θ) = -2qa.E.sin(θ), or
    τ = -p x E
     
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