Charge Is Distributed Uniformly Charge is distributed...

Charge Is Distributed Uniformly Charge is distributed uniformly throughout the volume of an infinitely long cylinder of radius $R$. (a) Show that, at a distance $r$ from the cylinder axis (for $r<R)$ $$ |\vec{E}|=\frac{|\rho| r}{2 \varepsilon_{0}} $$ where $|\rho|$ is the amount of volume charge density. (b) Write an expression for $|\vec{E}|$ when $r>R$

Charge Is Distributed Uniformly Charge is distributed uniformly throughout the volume of an infinitely long cylinder of
radius $R$. (a) Show that, at a distance $r$ from the cylinder axis (for $r<R)$
$$
|\vec{E}|=\frac{|\rho| r}{2 \varepsilon_{0}}
$$
where $|\rho|$ is the amount of volume charge density. (b) Write an expression for $|\vec{E}|$ when $r>R$

Key Concepts

Division of Regions (Inside and Outside the Cylinder)

When evaluating the electric field due to a uniformly charged cylinder, it is essential to consider separate regions: one inside the cylinder (r<R) and one outside the cylinder (r>R). The enclosed charge and the resulting electric field are treated differently in these regions. Inside the cylinder, the field increases linearly with radial distance, while outside the cylinder, it behaves as if all the charge were concentrated at the center, leading to an inverse relationship with the distance from the axis.

Uniform Charge Distribution

A uniform charge distribution means that the charge density is constant throughout the volume of the cylinder. This concept simplifies the analysis because the total charge enclosed within a given volume directly scales with that volume, and it ensures that the electric field inside and outside the cylinder can be calculated using simple algebraic relations derived from Gauss’s Law.

Gauss's Law

Gauss's Law, one of Maxwell’s equations, states that the total electric flux out of a closed surface is equal to the enclosed charge divided by the vacuum permittivity. This principle is fundamental in calculating electric fields generated by symmetric charge distributions, as it simplifies the computation by relating the flux through a Gaussian surface to the net charge within it.

Cylindrical Symmetry

Cylindrical symmetry is key when dealing with charge distributions over infinitely long cylinders, as it implies that the electric field at any point depends only on the radial distance from the cylinder’s axis, and not on the azimuthal angle or position along the length. This symmetry allows for the selection of a cylindrical Gaussian surface, making the integration and subsequent calculation of the electric field more straightforward.

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