5.3 Dipole in a Ratchet Electrical Potential
69
5.3 Dipole in a Ratchet Electrical Potential
Consider the following potential acting on the dipole (Fig. 5.2):
V rat (θ ) =
V 0
rat
2π
sin
2πθ
θ 0
−
1
4
sin
4πθ
θ 0
(5.22)
The corresponding electric field modulus at the position of the positive charge is:
E rat (θ ) = −
dV rat (θ )
dθ
= E
0
rat
cos
2πθ
θ 0
−
1
2
cos
4πθ
θ 0
(5.23)
where E 0
rat = −
V 0
rat
θ 0
. The corresponding electric field modulus at the position of the
negative charge is:
E rat (θ + π) = −E rat (θ )
(5.24)
Then the corresponding force modulus at the position of the positive charge is:
F rat (θ ) = qE rat (θ )
(5.25)
and the corresponding force modulus at the position of the negative charge is:
F rat (θ + π) = −qE rat (θ + π) = qE(θ)
(5.26)
We observe they are the same. Then the total ratchet torque acting on the dipole is:
E(rat)
p
(θ ) = 2qE rat (θ )
d
2
= pE rat (θ )
(5.27)
namely:
Fig. 5.2 Dipole under an
electric and ratchet field