Calculating E Field of Dipoles at Distance r: Griffiths 4.5 Solution

[Ghost117]

This is part of a question from Griffiths 4.5 (electrodynamics, 4th edition)

Homework Statement

p1 and p2 are (perfect) dipoles a distance r apart (Their alignment is such that p1 is perpendicular to the line separating them (pointing upwards) and p2 is parallel to the line separating them (pointing away from p1).)

What is Field of p1 at p2, and the Field of p2 at p1?

Homework Equations

E_dip(r,θ) = p/(4πε₀r³) * (2cosθr + sinθθ)

The Attempt at a Solution

I actually have the official solution for this, but I don't understand it... The theta value in the solution uses theta = π/2 for the field of p1 at p2, but a value of theta = π, for the equation for p2 at p1.. And I don't see where either of these theta values are coming from (my spherical coordinates are very weak, and I suspect that's the real problem here.)

Thanks

 

[TSny]

Note how θ is defined in the diagram (given back in chapter 3). See attached figure.
Imagine that the dipole shown is p₁. Where would p₂ be located in this figure? What would be the value of θ at the location of p₂?

 

[Ghost117]

Note how θ is defined in the diagram (given back in chapter 3). See attached figure.
Imagine that the dipole shown is p₁. Where would p₂ be located in this figure? What would be the value of θ at the location of p₂?

Yes that's the diagram I'm trying to use, and for E₁, I can see why theta = pi/2 (since p2 is on the y-axis in that diagram, pointing away.) But I don't understand why theta = pi when calculating E₂ (field of p2 at p1)... If I set p2 as my zero and count to p1 (clockwise) I get 3pi/2... not pi... I just don't see how I can get pi at all for any angle between these two dipoles...

 

[blue_leaf77]

When you want to calculate the field of $\mathbf{p}_2$ at $\mathbf{p}_1$, you have to use a new coordinate system in which the z axis is parallel to $\mathbf{p}_2$ and its positive direction points in the same direction as $\mathbf{p}_2$ does.

If I set p2 as my zero and count to p1 (clockwise) I get 3pi/2

Polar angle only runs from 0 to $\pi$.

 

[Ghost117]

When you want to calculate the field of $\mathbf{p}_2$ at $\mathbf{p}_1$, you have to use a new coordinate system in which the z axis is parallel to $\mathbf{p}_2$ and its positive direction points in the same direction as $\mathbf{p}_2$ does.

Polar angle only runs from 0 to $\pi$.

Thank you, I suspected it was going to come down to a basic problem with my understanding of the spherical coordinate system.