Long Metal Tube Figure 24-31 shows a section of a long,...

Long Metal Tube Figure $24-31$ shows a section of a long, thinwalled metal tube of radius $R$, with a positive charge per unit length $\lambda$ on its surface. Derive expressions for $|\vec{E}|$ in terms of the distance $r$ from the tube axis, considering both (a) $r>R$ and (b) $r<R$. Plot your results for the range $r=0$ to $r=5.0 \mathrm{~cm}$, assuming that $\lambda=2.0 \times$ $10^{-8} \mathrm{C} / \mathrm{m}$ and $R=3.0 \mathrm{~cm}$. (Hint: Use cylinderical Gaussian surfaces, coaxial with the metal tube.)

Long Metal Tube Figure $24-31$ shows a section of a long, thinwalled metal tube of radius $R$, with a positive charge per unit length $\lambda$ on its surface. Derive expressions for $|\vec{E}|$ in terms of the distance $r$ from the tube axis, considering both (a) $r>R$ and (b) $r<R$. Plot your results for the range $r=0$ to $r=5.0 \mathrm{~cm}$, assuming that $\lambda=2.0 \times$
$10^{-8} \mathrm{C} / \mathrm{m}$ and $R=3.0 \mathrm{~cm}$. (Hint:
Use cylinderical Gaussian surfaces, coaxial with the metal tube.)

Key Concepts

Gauss's Law

Gauss's Law states that the total electric flux through a closed surface is directly proportional to the charge enclosed within that surface. This principle is particularly powerful when applied to problems exhibiting high symmetry, as it can greatly simplify the calculation of electric fields by converting complex integrals into more manageable expressions.

Cylindrical Symmetry

Cylindrical symmetry occurs when the physical properties of a system are invariant under rotations about a particular axis. In the context of a long, thin-walled, charged tube, the electric field depends only on the radial distance from the axis, which simplifies both the analysis and calculation of the field.

Cylindrical Gaussian Surfaces

Cylindrical Gaussian surfaces are chosen to exploit cylindrical symmetry in problems involving elongated, cylindrical charge distributions. By aligning the Gaussian surface with the charged object, one can ensure that the electric field is either constant or zero along portions of the surface, thereby reducing the complexity of applying Gauss's Law.

Charge Distribution on Conductors

In conductors under electrostatic conditions, charges reside on the surfaces, and the electric field inside the conductor is zero. This characteristic significantly impacts the calculation of the electric field both inside and outside the conductor, leading to distinct expressions for each region based on the amount of enclosed charge.

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