Spherical Coordinates
Instructions: The animation above illustrates the geometry of the spherical coordinate system, showing its coordinate curves, surfaces, and basis vectors (explained below). Drag the screen to change the perspective of the camera. Drag the dot—or use the sliders—to change the position of the point. Use the checkboxes to control which elements are shown onscreen.
Explanation: Spherical coordinates $(r,\theta,\phi)$ are one of the most useful non-Cartesian systems of coordinates, and they’re applied frequently in physics problems. Instead of the usual three Cartesian coordinates $(x,y,z)$, we label the position of a point by its distance $r$ from the origin, its angle $\theta$ from the positive $z$ axis, and the angle $\phi$ from the positive $x$ axis to the “shadow” of the point in the $xy$ plane:
The coordinates are illustrated in the above figure; drag the dot or the sliders to see how the coordinates change from point to point.
We can visualize a coordinate system by drawing its coordinate curves, where we allow only one coordinate to vary at a time, holding the other two constant. In the case of Cartesian coordinates, the $x$, $y$, and $z$ coordinate curves are just straight lines that fill space with a rectangular grid, like the lines on a piece of graph paper:
In spherical coordinates, on the other hand, the analogous coordinate curves are shown in the figure at the top of the page. The $r$ coordinate curves—where we vary $r$ but hold $\theta$ and $\phi$ fixed—are radial lines emanating outward from the origin. Likewise, the $\theta$ coordinate curves are half circles beginning and ending on the $z$ axis, and the $\phi$ coordinate curves are circles wrapping all the way around the $z$ axis.
Watch how the coordinate curves change as you drag the dot to different points in space! In this way, we can visualize how the $r$, $\theta$, and $\phi$ coordinates change from point to point.
A coordinate system does more than just give us a set of labels $(r,\theta,\phi)$ to identify points in space, however. It also provides us with a set of basis vectors $\{\hat r, \hat \theta, \hat \phi\}$, that we can use to expand vectors like the velocity $\vec v$ of a particle, the force $\vec F$ acting on it, the electric field $\vec E$, and so on.
At any point in space, the $\hat r$ basis vector points along the $r$ coordinate curve at that point, the $\hat \theta$ basis vector points along the $\theta$ coordinate curve, and the $\hat \phi$ basis vector points along the $\phi$ coordinate curve. You can see the basis vectors by checking the “basis” checkbox in the figure at the top of the page.
Notice how the arrows rotate as you move the dot around to different points in space! Unlike the usual Cartesian basis vectors $\{\hat x, \hat y, \hat z\}$, the spherical basis vectors $\{\hat r, \hat \theta, \hat \phi\}$ change from point to point.
Finally, we come to the coordinate surfaces, which you can see by checking the “surfaces” checkbox. Remember that along, say, the $r$ coordinate curve, we let $r$ vary while we hold $\theta$ and $\phi$ fixed. By contrast, we can construct an $r$ coordinate surface by fixing $r$ to a constant while leaving $\theta$ and $\phi$ free to change.
Since $r$ measures the distance from the origin, a surface of constant $r$ is just a spherical shell surrounding the origin. Likewise, the $\theta$ coordinate surfaces are cones centered on the $z$ axis that open at that fixed angle $\theta$. And the $\phi$ coordinate surfaces are half-planes with one edge on the $z$ axis, oriented at the angle $\phi$ with respect to the $x$ axis.
It can be difficult to visualize all these 3D geometric objects in your head. I hope the above animations help!
To learn how to do physics in general coordinate systems—whether that’s spherical coordinates, cylindrical coordinates, or anything else—check out my series of in-depth online courses, Fundamentals of Tensor Calculus:
If you encounter any bugs in this animation, please let me know at feedback@PhysicsWithElliot.com.