Calculate the Capacitance of 3 conductors in the presence of the ground

[my_life]

Homework Statement

The conductors are cylindrical, equally spaced, and parallel to each other, with a radius r0 = 0.5 cm.
Conductors 2 and 3 are located at a height of h = 5 m from the ground; the distance between the conductors is equal to
d = 10 cm Let V1, V2, and V3 be the voltages applied to each conductor and Q1, Q2, and Q3 the charges for
a unit of length. Calculate the partial capacities of the system

Relevant Equations

C= Q/V0

Initially, I thought to apply the image charge theorem, but I don't know how to continue.
Someone could give me some indication on how to solve the problem? Thank you.

 

[Babadag]

Positive sequence capacitance:

2.π.εr.εo/ln(d/r) nF/km

Zero sequence capacitance:

2.π.εr.εo/{3.ln[2.h/(r.d^2)^(1/3)]} nF/km
d=(d12.d23.d31)^(1/3) [m]

h=(h1.h2.h3)^(1/3) [m]

For air ambient εr=1

εo =8.854 nF/km

r=conductor radius [m]

 

[Babadag]

I think you have another problem. Since they are given the potentials and the charges you have to employ this equation[typical]:
V12=1/(2.π.εr.εo).{Q1.[ln(d12/r)-ln(h12'/h1]+Q2.[-ln(d12/r)-ln(h2/h12')]+Q3.[ln(d23/d31)-ln(h23/h31)]}
Where V12=V1-V2
What do you need is V1,V2 and V3 if Q1=Q2=Q3=1?

 

[Babadag]

At the second look it seems that the calculated potential for each separate conductor is more useful.

V1=k.[Q1.ln(h1/r)+Q2.ln(h12/d12)+Q3.ln(h13/d13)]

V2=k.[Q2.ln(h2/r)+Q1.ln(h12/d12)+Q3.ln(h23/d23)

V3=k.[Q3.ln(h3/r)+Q1.ln(h13/d13)+Q2.ln(h23/d23)]

k=1/(2.π.εr.εo)= 17975484.88 km/F

If Q=1C[Coulomb] V1[V] Cap=1C/V[F]