Final answer:
Cylindrical symmetry holds when the electric field is the same at all points that have the same distance from the center of the object. The situations described in the question involve different rods and shells with charges and distances given, and we need to determine if the electric field at different points is symmetric or not.
Step-by-step explanation:
The given question is asking about cylindrical symmetry in different situations involving charged rods and shells. Cylindrical symmetry holds when the electric field of a charged object is the same at all points that have the same distance from the center of the object. In each situation described, we need to determine if the electric field at different points is the same, and if not, why it is not symmetric.
(a) A 300-cm long copper rod with a charge of +500 nC and a point 5 cm from its center is not cylindrically symmetric because the electric field will depend on the position along the rod.
(b) A 10-cm long copper rod with a charge of +500 nC and a point 5 cm from its center is cylindrically symmetric because the electric field will be the same at all points that have the same distance from the rod's center.
(c) The wooden and plastic rods glued together to form a 300-cm long rod with a uniform charge density and a point 4 cm from its center is cylindrically symmetric because the electric field will be the same at all points that have the same distance from the rod's center.
(d) The situations involving the copper shell and aluminum sphere, with charges and distances given, will depend on the exact geometry of the setup and can be analyzed accordingly.