A disk of radius R (Fig. P25. 71) has a nonuniform surface...

A disk of radius $R$ (Fig. P25. 71$)$ has a nonuniform surface charge density $\sigma=C r,$ where $C$ is a constant and $r$ is measured from the center of the disk. Find (by direct integration) the potential at $P .$

Key Concepts

Differential Charge Element

A differential charge element, dq, is an infinitesimal portion of the total charge located on a small area element of the surface. It is expressed as dq = ? dA, where dA is the differential area element. In polar coordinates, which are well-suited for circular geometries, dA is typically written as r dr d?, allowing for the integration to account for the circular symmetry of the problem.

Polar Coordinate Integration

Using polar coordinates is a natural choice for solving problems involving circular or disk-shaped charge distributions due to their inherent symmetry. In polar coordinates, the area element dA becomes r dr d?, and the limits of integration are set according to the radius of the disk and the full angular span. This coordinate transformation simplifies the integration process and helps in handling nonuniform charge distributions by aligning the coordinate system with the symmetry of the problem.

Electric Potential

The electric potential due to a charge distribution is evaluated by integrating the contributions of all infinitesimal charge elements. Each contribution is given by the expression dV = dq/(4???r), where dq is an element of charge and r is the distance from the charge element to the point of interest. Calculating the potential often involves setting up an integral over the spatial extent of the distribution with proper limits and coordinate system to account for the geometry of the problem.

Surface Charge Density

Surface charge density represents the amount of charge per unit area distributed on a surface. In cases where the charge density is not uniform, it becomes a function of position (? = C r in this example). This spatial dependence requires modifying the integration process, as the amount of charge in each differential area element varies with position, necessitating careful inclusion of the functional form in the integrals.

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