Question

What is an infinitely long cylinder? When is it proper to treat an actual cylinder as being infinitely long, and when is it not?

Short answer

Answer: It is appropriate to treat a cylinder as infinitely long when the length of the cylinder is significantly larger than the other dimensions and the problem involves effects or phenomena that occur far from the ends of the cylinder. In these cases, end effects can be neglected. On the other hand, it is not appropriate to assume an infinitely long cylinder when the length of the cylinder is comparable to or smaller than the other dimensions, or when the problem explicitly involves the interaction between the ends of the cylinder and the surrounding environment. The decision to treat a cylinder as infinitely long depends on the context of the problem and the degree of accuracy required.

Step-by-step solution

Definition of an Infinitely Long Cylinder

An infinitely long cylinder is a cylinder that has an infinite length along its central axis while maintaining a constant cross-sectional shape and area. In other words, its two circular bases are of the same size and it extends indefinitely in both directions along the axis.

Use of the Infinitely Long Cylinder Approximation

It is proper to treat an actual cylinder as being infinitely long when the length of the cylinder is significantly larger than the other dimensions (i.e., the diameter) and the problem involves effects or phenomena that occur far from the ends of the cylinder. In such a case, the main fields or quantities of interest are primarily determined by the cylindrical geometry, and the end effects can be neglected.

When the Approximation is not Valid

On the other hand, it is not appropriate to assume an infinitely long cylinder when the length of the cylinder is comparable to or smaller than the other dimensions, or when the problem explicitly involves the interaction between the ends of the cylinder and the surrounding environment (e.g., fluid flow around the cylinder, heat transfer between the cylinder ends, and the medium).

Context-dependent Decision

In conclusion, the decision to treat a cylinder as infinitely long depends on the context of the problem being solved and the degree of accuracy required. In general, if it can be reasonably assumed that the end effects will not significantly influence the outcome and if the length is substantially larger than the other dimensions, then the approximation of an infinitely long cylinder can be used to simplify calculations.