A rod of length L (Fig. P 24.25 ) lies along the x axis with...

A rod of length $L$ (Fig. $\mathrm{P} 24.25$ ) lies along the $x$ axis with its left end at the origin. It has a nonuniform charge density $\lambda=\alpha x,$ where $\alpha$ is a positive constant. (a) What are the units of $\alpha$ ? (b) Calculate the electric potential at $A$.

Key Concepts

Integration of Continuous Charge Distribution

For calculating the electric potential due to a continuous charge distribution, the method involves integrating the contributions of infinitesimal charge elements. This integration, based on Coulomb’s law, accounts for variations in the charge density and the geometry of the charge distribution, thereby providing the net potential at a point.

Dimensional Analysis

Dimensional analysis is used to ensure that equations are physically consistent by matching units on both sides. In the context of a charge density expression, it helps determine the units of constants so that the overall units of the charge density are maintained (in this case, Coulombs per meter).

Charge Density

Charge density represents the amount of electric charge distributed per unit length, area, or volume. When the charge density is given as a function of position (such as a linear variation along a rod), it signifies that the charge is not distributed uniformly, affecting subsequent calculations of electrical quantities.

Electric Potential

Electric potential is the work done per unit charge in bringing a test charge from infinity to a point in space. It is a scalar quantity obtained by summing (or integrating) the contributions from all charge elements of a distribution, which is particularly important when dealing with nonuniform charge distributions.

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