Section 1.7: Integration to Find Forces
We use calculus to compute the net force exerted by continuous charge distributions on a point charge by integrating infinitesimal force contributions.
Coulomb’s Law for Continuous Charges:
\( d\vec{F} = k_e \frac{dq \, q_0}{r^2} \hat{r} \),
\( \vec{F} = k_e q_0 \int \frac{dq}{r^2} \hat{r} \)
Here, \( dq = \lambda dl, \sigma dA, \rho dV \) depending on geometry.
Example 1
A rod of length L carries a uniform linear charge density \( \lambda \). Find the force on a point charge \( q_0 \) located along the perpendicular bisector of the rod.
Use \( dq = \lambda dx \), integrate: \( \vec{F} = k_e q_0 \int_{-L/2}^{L/2} \frac{\lambda dx}{(x^2 + d^2)} \hat{r} \). Evaluate the integral to find magnitude and direction.
Practice Problems
- A ring of radius R carries total charge Q. Find the force on a charge q located at the center.
- A disk of radius R has uniform surface charge density σ. Calculate the net force on a point charge q above the center.
- Determine the net force on a point charge due to a uniformly charged rod along its axis.
- Explain how to set up the integral for a 3D charge distribution (volume charge density ρ).
- Compute the force on a point charge due to a semicircular arc with linear charge density λ.