Question

A vertical infinite wire along the z-axis has two symmetries: One is a rotational symmetry around the z-axis while the other is a translational symmetry along the z-axis. Describe the symmetries of a) a sphere, b) an infinite plane, c) a circular plate, d) a finite length rod with circular cross section, and e) a finite length bar with a square cross section.

Please explain process.

Answer 1

a) Sphere.

Position the center of the sphere at origin of coordinate axes. Then the continuous symmetries are:

Rotation about X axis

Rotation about Y axis

Rotation about Z axis.

In fact the sphere will be symmetric under rotation about any axis passing through the origin, but those rotations can be written in terms of rotations about X,Y and Z axis.

Note that continuous symmetry means, you can rotate by ANY angle and the configuration will still be symmetric.

b) Infinite plane.

Suppose the plane is in X-Y plane. The Z axis is perpendicular to the plane.

The continuous symmetries are:

Rotation about Z axis.

Translation along X direction

Translation along Y direction.

In fact translation along any direction on the XY plane will keep the plane symmetric but those translations can be written in terms of translations along X and Y directions.

c) circular plate.

Suppose the plate is in X-Y plane. The Z axis is perpendicular to the plane.

If the radius is finite then there is one continuous symmetry:

Rotation about Z axis.

d) A finite length rod with circular cross section

Suppose the vertical axis of the rod is the Z axis. Then the continuous symmetry is:

Rotation about Z axis.

It doesn't have translation symmetry because the length is finite.

e) A finite length bar with a square cross section.

Suppose the vertical axis of the rod is the Z axis. Then it does NOT have any CONTINUOUS symmetry.

However, there is a discrete symmetry:

Rotation about Z axis by ONLY 90 degrees or multiple of 90 degrees. Because if you rotate a square by 90 degrees, it will come back to its original configuration.