A thin rod of length L lies on the x-axis with its center at...

A thin rod of length $L$ lies on the $x$-axis with its center at the origin, as shown in Fig. 20.29. The rod carries charge $Q$ distributed uniformly over its length. (a) Find an expression for the electric field at point $B$ in Fig. $20.29$, located on the positive $x$-axis a distance $x$ from the origin, where $x>L / 2$, so that point $B$ is beyond the right end of the rod. (b) Show that your result reduces to the field of a point charge $Q$ when $x \gg L$.

Key Concepts

Continuous Charge Distributions

This concept involves treating an extended charged object as a collection of infinitesimal charge elements, each contributing a small electric field that is summed via integration. Rather than applying Coulomb’s law to a point charge, the object’s charge is segmented and the total electric field is obtained by integrating the contributions from all these segments over the entire distribution.

Uniform Linear Charge Density

When a charged rod carries a constant total charge distributed uniformly along its length, the charge per unit length is the same at every point. This simplifies the problem by allowing the use of a constant density in the integral, which in turn leads to straightforward expressions for the electric field after summing the effects of all the differential charge elements.

Integration Techniques in Electrostatics

Calculating the electric field due to an extended charge distribution requires setting up an integral where the contribution from each differential charge element is computed using Coulomb’s law. Attention to the geometry and symmetry of the problem (such as the position of the rod along the x-axis) is crucial for determining the limits of integration and simplifying the resulting expression.

Far-field Approximation

In the far-field region, where the observation point is much farther from the charge distribution than the size of the distribution, the detailed structure of the source becomes less relevant. The charge distribution effectively behaves like a point charge, allowing the electric field to be approximated by the simpler expression for a point charge, which greatly simplifies analysis and calculation.

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