Magnetic field due to magnetic dipole

[fricke]

It is NOT a homework question. I am doing my revision and get stuck at this question.

I am confused with the angle θ shown in this link:
http://www.physicspages.com/2013/10/06/mutual-inductance/

Professor who wrote this solution stated that θ is the angle between unit vector z and unit vector r, but as what i understand, the angle between two unit vectors (in this question) is cos(180-θ) = -cos(θ) but it is different with the solution shown in the link. (Help me, I am so confused!)

And also one more question. Why is the integration of area taken from 0 to b? If it is, aren't we taking a integration of disk, then? but in the question, it says it is a loop.

Can someone please explain it briefly?

 

[Dale]

Professor who wrote this solution stated that θ is the angle between unit vector z and unit vector r, but as what i understand, the angle between two unit vectors (in this question) is cos(180-θ) = -cos(θ) but it is different with the solution shown in the link. (Help me, I am so confused!)

The professor is simply using the geometric definition of the dot product:
$\vec{a} \cdot \vec{b} = ||\vec{a}||\;||\vec{b}|| \; \cos(\theta)$
See: https://en.wikipedia.org/wiki/Dot_product#Geometric_definition

Since $\hat{z}$ and $\hat{r}$ are both unit vectors their magnitudes are both 1 and so $\hat{z} \cdot \hat{r} = \cos(\theta)$.

 

[fricke]

The professor is simply using the geometric definition of the dot product:
$\vec{a} \cdot \vec{b} = ||\vec{a}||\;||\vec{b}|| \; \cos(\theta)$
See: https://en.wikipedia.org/wiki/Dot_product#Geometric_definition

Since $\hat{z}$ and $\hat{r}$ are both unit vectors their magnitudes are both 1 and so $\hat{z} \cdot \hat{r} = \cos(\theta)$.

Thank you very much for your reply!
I still want to make sure one more thing here.
So, θ is the angle between these two unit vectors? or angle 180 degree subtract with the angle between unit vectors?

 

[jtbell]

So, θ is the angle between these two unit vectors?

Yes.