Electrostatic potential due to a dipole

[Crystal037]

Homework Statement

Find the electrostatic potential due to a dipole

Relevant Equations

V=Q/4*pi*ebselon

Given here is that by geometry
r1^2 =r^2 +a^2 - 2ar*cos(theta)
But if we try to do vector addition then since direction of dipole is upwards then it should be
r^2 =r1^2 +a^2 + 2ar1*cos(alpha)
Where alpha is the angle between a and r1. I Don,'t understand how they get it by geometry

 

[haruspex]

if we try to do vector addition then since direction of dipole is upwards then it should be
r^2 =r1^2 +a^2 + 2ar1*cos(alpha)

What vectors are you adding to get that? What has the geometry of a triangle got to do with the direction of a field?

 

[Crystal037]

Then why at one equation a negative sign is present and on the other equation a positive sign is present in front of the term 2racos(theta)

 

[haruspex]

Then why at one equation a negative sign is present and on the other equation a positive sign is present in front of the term 2racos(theta)

I cannot answer that until you explain how you get the equation with the positive sign.

 

[Crystal037]

I didn't got it. In the book it's simply written that we got it by geometry

 

[gleem]

cos(α) = -cos(180° -α)

 

[haruspex]

I didn't got it. In the book it's simply written that we got it by geometry

You don't seem to be understanding my question.
The book gives this equation, with a negative sign, by geometry, and that is quite easily got by the cosine rule:

r1² =r² +a² - 2ar*cos(theta)

Then you wrote this equation with a positive sign, and do not explain how you get it:

But if we try to do vector addition then since direction of dipole is upwards then it should be
r² =r1² +a² + 2ar1*cos(alpha)

What is alpha? I don't see that in the diagram. Did you mean theta?
If you take the line of length 2a and continue that upwards, then label the angle between that and the line length r1 as alpha then both equations are correct.

 

[Crystal037]

I have taken alpha as the angle between a and r1. But I was not talking about my equation. In the book itself there are 2equations, if you see the image then you'll see that after they have written by geometry they have written 2 equations:
First is r1^2 =r^2 +a^2 - 2ar cos(theta)
Second is r2^2 =r^2 +a^2 +2arcos(theta)
Now I understood that they have taken cosine rule and cos(180-theta) =-cos(theta)