Derive an equation for electric potential V(z) on z axis

[jackMybrain@ru]

Homework Statement

Hi, I am stuck with this homework. I have been asked to make an equation out of a diagram for V(z) using V=kQ/r equation, where z is a positive axis centred with four negative charges. Here is the diagram.

2. Relevant equation
E=F/q
F of point charge: F=kQq/r^2
E=kQ/r^2
V=kQq/r ~U=V*q & KE=0.5m*velocity^2

r=distant of the point from the point charge q
k=1/4*pi*[E][/0]

The Attempt at a Solution

I have already completed the (a) and (b) parts. However, for part (b) I could only make V(z)=Q/(4*pi*Enod)*r where r = 0.5*sqrt([d^2+(2Z-d)^2]) from the r12. I found r by taking all point charges distant together by saying r12 of the first 2 charges are equal and r34 of the 3rd and 4th is same. Therefore, Vtotal=KQ*[(2/r12)+(2/r34)]. But how do I make it as V(z)=-Q/pi*Enod... because currently I am getting V(z)=-Q/8*pi*Enod(sqrt([d^2+(2Z-d)^2])).
Am I doing something wrong. Thanks for any insight or solution for the other parts would be very helpful.
And how do I do Q 1. (c),(d)& (f)??

 

[jackMybrain@ru]

Homework Statement

Hi, I am stuck with this homework. I have been asked to make an equation out of a diagram for V(z) using V=kQ/r equation, where z point start in the center with four negative charges. Here is the diagram for the aid.

2. Relevant equation
E=F/q
Force of point charges by Coulumbs law: F=kQq/r^2
E=kQ/r^2
V=kQq/r ~U=V*q & KE=0.5*mass*velocity^2

r=distant of the point from the point charge q
k=1/4*pi*Enod

The Attempt at a Solution

I have already completed the (a) and (b) parts.
However, for part (b) I don't understand how to make a negative result for V(z)??
As normal equation goes, I have derived only V(z)=Q/(4*pi*Enod)*r where r = 0.5*sqrt([d^2+(2Z-d)^2]) (from the r12).
When I put everything together my found equation: V(z)=-Q/8*pi*Enod(sqrt([d^2+(2Z-d)^2])).
But how do I make it as V(z)=-Q/pi*Enod.(...) because both distance and z axis are positive.

**I found my d or r by taking all point charges distant from z together. For instance accumulating r12 of q1 and 12 are equal apart from z on z axis and r34 of the 3rd and 4th are higher than them vice versa. Therefore, (r) total=(2/r12)+(2/r34). So, Vtotal=KQ/[(2/r12)+(2/r34)].

Am I doing something wrong. I am only trying to understand physics. Thanks for any insight or solution for the other parts would be very helpful.
And how do I do Q 1. (c),(d)& (f)??

This is an easier version of my question. Please, help! I don't know why nobody is familiar with V(z) equation or how to solve them. Because there are not many references or videos to be found by online searching. Thanks again..

 

[ehild]

Show your derivation for r12 and r34 in detail.