Section 1.5: Force in Continuous Distributions
For a continuous charge distribution, the net force on a test charge is found by integrating infinitesimal contributions \( d\vec{F} \) from each element of charge \( dq \).
Force Calculation:
\( \vec{F} = k_e q \int \frac{d\vec{q}}{r^2} \hat{r} \)
Here \( dq \) is the infinitesimal charge element, and \( \hat{r} \) is the unit vector from \( dq \) to the test charge.
Example 1
A uniformly charged rod of length L exerts force on a point charge at one end. Find the net force using integration.
Set up \( dq = \lambda dx \) and integrate along the rod: \( \vec{F} = k_e q \int_0^L \frac{\lambda dx}{x^2} \hat{i} \).
Practice Problems
- Find the force on a point charge near a uniformly charged ring.
- Compute the net force from a uniformly charged disk on a point along its axis.
- Linear charge distribution along a line segment: find force at a perpendicular distance.
- Continuous sheet of charge: force at a point above center.
- Combine superposition and integration: multiple rods in different orientations.