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 an actual cylinder as infinitely long when its length is much greater than its cross-sectional dimensions (like radius) and the physical processes or properties depend mainly on this length, such as electric fields around very long charged wires or magnetic fields around long current-carrying conductors. It is not appropriate when the cylinder's length is comparable to or smaller than its cross-sectional dimensions or when the physical processes are critically influenced by its finite length, such as in Stokes' law problem or heat conduction through a short cylinder.

Step-by-step solution

Define an Infinitely Long Cylinder

An infinitely long cylinder is a hypothetical geometric object that extends infinitely in length (in one direction), while having a finite cross-sectional area. It is a theoretical concept and not a real object, as creating an infinitely large structure isn't physically possible. However, it is still useful for illustrating some principles in physics and mathematics, such as electrostatic and magnetic fields around a long charged conductive wire or fluid dynamics around a cylindrical object.

When to Treat a Cylinder as Infinitely Long

A real cylinder can be treated as an infinitely long cylinder when its length is much greater than its cross-sectional dimension (like radius) and when the physical processes or properties of the cylinder depend mainly on this length. This approximation serves to simplify the analysis of the problem or system under consideration. Some examples where this approximation is relevant include: 1. The electric field around an infinitely long, uniformly-charged wire can be analyzed as an infinitely long cylinder to approximate the behavior of very long wires. 2. The magnetic field around a straight current-carrying conductor can be approximated as an infinitely long cylinder if the conductor is multiple times longer than its radius.

When Not to Treat a Cylinder as Infinitely Long

An actual cylinder should not be treated as infinitely long when its length is comparable to or smaller than its cross-sectional dimensions (like radius), or when the physical processes or properties of the cylinder are critically influenced by the finite length of the cylinder. Some examples where this approximation is not relevant include: 1. In the Stokes' law problem considering the motion of a sphere through a viscous medium, the theory for an infinitely long cylinder will not be applicable as the sphere's size is large compared to the length of the cylinder and the physical principles are quite different. 2. For heat conduction through a cylinder, if the aspect ratio (length:radius) of the cylinder isn't large, then treating the cylinder as infinitely long is not appropriate, as heat transfer dynamics will be significantly different due to the temperature gradient and the finite size of the cylinder.

Key concepts

Electrostatic Fields

Electrostatic fields are a fundamental concept in physics that describe the force field surrounding charged particles. Imagine you have a charged balloon, and you bring it close to small pieces of paper. The way the paper moves towards the balloon can be understood through electrostatic fields. These fields influence other charges in the vicinity, resulting in attractive or repulsive forces.

In the context of an infinitely long cylinder, we use this principle when dealing with long, charged objects such as wires. The electric field around a wire can be complex, but assuming the wire is infinitely long allows us to simplify the calculations. The field can then be described by using Gauss's Law, which relates the electric field to the charge distribution. For an infinitely long and uniformly charged cylinder, the electric field outside the cylinder can be calculated by considering a cylindrical Gaussian surface that encircles the wire. The field inside is zero if it's a conductor, due to charge redistribution.

However, it's essential to remember that this is an idealization. In practical situations, when the wire isn't significantly longer than its diameter, edge effects become significant, and the field can no longer be treated as if originating from an infinitely long source.

Magnetic Fields

Magnetic fields can be envisioned as the influence that magnets and electric currents exert in the space around them, causing materials like iron filings to align along the field lines. The behavior of magnetic fields around currents and magnets is central to the operation of a wide variety of devices, from electric motors to MRI machines.

Regarding an infinitely long cylinder, we're often talking about a straight wire carrying a steady current. When we assume the wire is infinitely long, we can apply Ampere's Law to find the magnetic field around the wire. Ampere's Law helps us understand that the magnetic field around a straight wire is proportional to the electric current and inversely proportional to the distance from the wire.

This simplification assists in designing and understanding long transmission lines or the coils of electromagnets, where the ends of the wire play a negligible role in the overall behavior of the magnetic field. Still, if the wire or cylinder has a finite length, we need to consider the edge effects and cannot purely rely on the simplifications made for an infinitely long approximation.

Heat and Mass Transfer

Heat and mass transfer comprises the movement of thermal energy and various substances from one place to another. These can range from the heating of your coffee mug to the dispersion of a pollutant in a river. Heat transfer can occur through conduction, convection, or radiation, while mass transfer often involves diffusion or convection.

In the hypothetical scenario of an infinitely long cylinder, if we consider heat transfer, the analysis simplifies as we neglect the ends of the cylinder. This is because the length is so much greater than any other dimension that the temperature profile becomes effectively one-dimensional, along the radial direction from the cylinder's surface to its center. The principles of Fourier's law for heat conduction apply here, allowing for the derivation of temperature fields.

However, this assumes that the ends of the cylinder do not contribute to the heat transfer, which is not the case in finite, real-world objects. Here, boundaries play a significant role, and the analysis must be adjusted to consider the three-dimensional nature of the temperature fields and the various modes of heat transfer.

Fluid Dynamics

Fluid dynamics is the study of fluids (liquids and gases) in motion. It encompasses complex behaviors as diverse as the swirling of cream in your morning coffee and the flow of air over an airplane wing. The fundamental principles of fluid dynamics are based on conservation laws, including the conservation of mass (continuity equation) and momentum (Navier-Stokes equations).

The concept of an infinitely long cylinder in fluid dynamics helps us tackle problems like flow around cylindrical structures such as pipes and cables. Assuming the structure's length infinitely long excludes end effects, allowing us to focus on the flow pattern around the cylinder's curved surface. For instance, the flow around a long pipe submerged in a river can be approximated using this model, leading to solutions that predict the distribution of pressure and velocity of the fluid.

However, just as with the other fields discussed, this is a simplification that neglects several real-world factors. Should the cylinder be finite in length, flow separation and turbulent wake regions at the ends of the cylinder become vital to account for. Therefore, the fluid dynamics around such a cylinder requires a more complex approach to capture the three-dimensional nature of the flow, particularly near the ends.