Spherical coordinates - phi vs theta

[Calpalned]

Homework Statement

My textbook states that when $\theta = c$ where c is the constant angle with respect to the x-axis, the graph is a "half-plane". However, when $\phi = c$ it is a half-cone. The only difference I see is that $\phi$ is the angle with respect to the z-axis, rather than the x-axis. Why is it that one ends up being a cone and the other a plane?

Homework Equations

n/a

The Attempt at a Solution

That is the only part I don't understand. It is clear how $\rho = c$ is a sphere, but why are $\phi$ and $\theta$ so different?

 

[LCKurtz]

Homework Statement

My textbook states that when $\theta = c$ where c is the constant angle with respect to the x-axis, the graph is a "half-plane". However, when $\phi = c$ it is a half-cone. The only difference I see is that $\phi$ is the angle with respect to the z-axis, rather than the x-axis. Why is it that one ends up being a cone and the other a plane?

Homework Equations

n/a

The Attempt at a Solution

That is the only part I don't understand. It is clear how $\rho = c$ is a sphere, but why are $\phi$ and $\theta$ so different?

Look at the geometry. If you pick a point where $\theta = c$ and vary $r>0$ and $\phi$ do you see why you get a half plane? Now pick a point where $\phi = c$. What happens to that point as you vary $\theta$? When you very $r$? It might help to be looking at http://en.wikipedia.org/wiki/Spherical_coordinate_system when you answer. Use the second picture there as it has the usual notation used in math.

 

[HallsofIvy]

Homework Statement

My textbook states that when $\theta = c$ where c is the constant angle with respect to the x-axis, the graph is a "half-plane". However, when $\phi = c$ it is a half-cone. The only difference I see is that $\phi$ is the angle with respect to the z-axis, rather than the x-axis.

You are mistaken. For one thing, $\theta$ can range from 0 to $2\pi$ while $\phi$ only ranges from 0 to $\pi$. Do you see why that is true?
("Mathematics notation" and "physics notation" reverse $\theta$ and $\phi$. I am assuming that you are using "mathematics notation".)

Why is it that one ends up being a cone and the other a plane?

Homework Equations

n/a

The Attempt at a Solution

That is the only part I don't understand. It is clear how $\rho = c$ is a sphere, but why are $\phi$ and $\theta$ so different?

 

[BvU]

How about some physical experience ?

Stand up straight and lift one arm straight up. Let the direction your feet are pointing in represent $\theta$ and the angle your arm makes with the vertical when you lower it represent $\phi$ ("math notation" - I didn't know it existed). Fortunately they have r in common imagine a very long arm...

Moving your arm up and down ($[0,\pi]$) with feet fixed defines a half plane.
And rotating on your feet ($[0,2\pi]$) with arm angle fixed gives you the cone. (I take it the mistake Ivy refers to is that it's a cone surface, not half a cone).

And now I will revert to "physics notation" in order not to confuse myself -- after all this is PF and not MF !