Topic B Flashcards
What is an electric dipole?
Pair of equal & opposite charges, Q & –Q, separated
by distance s
What is an ideal dipole?
s (separated distance between charges) is very small compared to the distances to any other charges
How do you find the resultant electric potential to any point?
Assume V=0 at infinity and the dipole is ideal(s
What is the dipole moment?
vector p=Qs
where s is defined as pointing from -Q to +Q
How do you find the resultant electric field to any point?
To find E we use E=-∇V but for convenience, we use the cylindrical form of ∇[due to the symmetry of the system].
E_r =∂V/∂r =2pcosθ/4πε_0r^3
and
E_θ =∂v/∂θ*1/r=psinθ/4πε_0r^3
The total E-field has a magnitude
E=sqrt(E_r^2+E_θ^2)=p(1+3cos^2θ)^1/2/4πε_0r^3
[THE DIRECTION OF E_r AND E_θ CHANGES AS θ CHANGES i.e if θ increases E_r becomes vertical but E_θ remains perpendocular]
What is the angle ϕ between the total electric field and the electric field to r ?
tanϕ=E_θ/E_r=1/2tanθ
What happens when a dipole is in a uniform field?
When a dipole, where the distance d between the charges is fixed, is placed in a uniform electric field E it experiences no net force, as the two charges feel equal and opposite forces(+EQ&-EQ) but there is the torque
How does torque arise for dipoles in a uniform electric field?
the two forces do not act through the same point, there is a torque acting
on the dipole
Torque = Force x perpendicular distance
What is the torque for dipoles in a uniform electric field?
T= pEsinθ
torque is a vector quantity whose direction gives the axis of rotation which is normal to
the plane containing p and E
T=pxE(cross product)
The direction of T is given by the right hand screw rule
How does potential energy arise for a dipole in a uniform electric field?
The dipole tends to align itself with an applied external field. This means that it takes work (provided by an opposing torque or pair of forces) to rotate the dipole against
the field, resulting in an increasing potential energy of the dipole. D
What is the potential energy of dipoles in a uniform electric field?
- Choose zero energy configuration when both charges are on the same equipotential line so total energy is zero
- Calculate the work done in rotating dipole from zero potential position to the new position
Work done = Torque x angle = ∫Tdθ
U = −pE cos θ = −p · E
What is the force on a dipole in a non-uniform E-field?
F_x=pdE_x/dx
Force is proportional to rate at which field changes with distance – this is a general result for
any orientation of the dipole
What is the torque in a non-uniform field?
y T=pxE unless the E-field varies significantly
over the spatial extent of the dipole (in this case we would not have an ideal dipole)
What are quadrupoles?
Where sum of dipole moments equals zero, multiple dipoles in one system.
In analogy to single charges (monopoles) and dipoles the potential due to a quadrupole falls
off as r^-3 and the E-field as r^-4
