Frank Einstein
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- 1
Homework Statement
In the XY plane there are two point charges +q at (0, -b, 0) and –q at (0, b, 0) and a ring of radius a centred in the centre of the plane:
Find the electric field in all points of the z axis and study the field’s dominant behaviour at distances z >>a,b.
Find the electric field at any point of space at big distances by using the two firsts terms of the multipolar development and compare with the previousparagraph.
The z axis pops out of the paper and goes up.
Homework Equations
All bold leters are vectors
any x^, y^, z^, is a vector of module 1.
p= sum(ri*qi)
Φ1=(1/4*π*ε0)*(r*p/r^3)
The Attempt at a Solution
Well, I don’t have much trouble with the first part, I find the field of a positive punctual charge at the z axis, same with a negative and by superposing both, I finish with
–b/2*π*ε0*(z^2+b^2)^0.5 in the y direction.
For the ring, I have (λ*a)/(2*ε0*(z^2+a^2)^(3/2) in the z direction. λ is the linear density of charge
At big distances the total field is -b/(2*π*ε0*z)y^+ (λ*a)/(2*ε0*z^2)z^Trouble comes when I arrive to the second part because when I calculate the three contributions the monopolar, dipolar and quadrupolar. The ring doesn’t produce any multipolar development
The first one is 0, the total charge is 0, the second contribution, the dipolar one is pr/(4*π*ε0), p=2bqy^
And last, when I have to find the quadripolar, momentum as
(¼*π*ε0)*Σ(Qij*(3xixj-(r^2)*δij)/r^5)
I find that Q11=Qxx=0, Qzz=Q33=0 and Q22=Qyy=(1/2)[(-b)(-b)q+(b)(b)(-q)]=0.
Meaning that the quadripolar momentum is 0, I know that I cannot go to further terms because the teacher has said that we won’t study these, so I find myself with just one term when in the description of the problem they tell me to use two terms.
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