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

1. Apr 8, 2012

### Jonnie79

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. Apr 9, 2012

### Jonnie79

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