AP Physics C Questions on Electric Potential due to Continuous Charge Distributions

📚 Quick Study Guide

• 📏 [Dimensionality Matters!] Electric potential calculations depend on the charge distribution's geometry (1D, 2D, or 3D).
• ➕ [Superposition Principle] The total electric potential at a point is the scalar sum of the potentials due to each infinitesimal charge element: $V = \int dV = \int \frac{kdq}{r}$.
• 💡 [Linear Charge Density] For a line of charge: $\lambda = \frac{Q}{L}$, where Q is the total charge and L is the length.
• ⚡ [Surface Charge Density] For a surface of charge: $\sigma = \frac{Q}{A}$, where Q is the total charge and A is the area.
• ∭ [Volume Charge Density] For a volume of charge: $\rho = \frac{Q}{V}$, where Q is the total charge and V is the volume.
• 🧮 [Symmetry is Key!] Exploit symmetry to simplify the integral. Choose your coordinate system wisely!
• 🛡️ [Potential is Scalar] Remember, electric potential is a scalar quantity, so you only need to integrate the magnitude, not the direction.

🧪 Practice Quiz

1. A thin rod of length $L$ has a uniform charge density $\lambda$. What is the electric potential at a point $P$ located along the axis of the rod, a distance $d$ from one end?
   1. $V = k\lambda \ln(\frac{L+d}{d})$
   2. $V = k\lambda \ln(\frac{L}{d})$
   3. $V = k\lambda (L+d)$
   4. $V = k\lambda d$
2. A ring of radius $R$ has a uniform charge $Q$. What is the electric potential at a point on the axis of the ring, a distance $x$ from the center?
   1. $V = \frac{kQ}{\sqrt{R^2 + x^2}}$
   2. $V = \frac{kQ}{R+x}$
   3. $V = \frac{kQ}{R}$
   4. $V = \frac{kQ}{x}$
3. A uniformly charged disk of radius $R$ has a surface charge density $\sigma$. What is the electric potential at a point on the axis of the disk, a distance $x$ from the center?
   1. $V = 2\pi k \sigma (\sqrt{R^2 + x^2} - |x|)$
   2. $V = 2\pi k \sigma R$
   3. $V = 2\pi k \sigma x$
   4. $V = 2\pi k \sigma (R - x)$
4. Two large, parallel, conducting plates are separated by a small distance $d$. The plates carry equal and opposite surface charge densities of magnitude $\sigma$. What is the potential difference between the plates?
   1. $V = \frac{\sigma d}{\epsilon_0}$
   2. $V = \frac{\sigma}{\epsilon_0 d}$
   3. $V = \sigma d \epsilon_0$
   4. $V = \frac{d}{\sigma \epsilon_0}$
5. A sphere of radius $R$ has a uniform volume charge density $\rho$. What is the electric potential at the surface of the sphere?
   1. $V = \frac{kQ}{R}$
   2. $V = \frac{3kQ}{2R}$
   3. $V = \frac{kQ}{2R}$
   4. $V = \frac{2kQ}{3R}$
6. A long, non-conducting cylinder of radius $R$ has a uniform volume charge density $\rho$. What is the electric potential at a distance $r > R$ from the axis of the cylinder?
   1. $V(r) = \frac{\lambda}{2\pi\epsilon_0} \ln(\frac{R}{r})$ where $\lambda$ is the charge per unit length.
   2. $V(r) = \frac{\lambda}{2\pi\epsilon_0} r$
   3. $V(r) = \frac{\lambda}{2\pi\epsilon_0} R$
   4. $V(r) = \frac{\lambda}{2\pi\epsilon_0} \ln(r)$
7. A point charge $+Q$ is located at the center of a conducting spherical shell of inner radius $a$ and outer radius $b$. What is the electric potential at a point outside the shell ($r > b$)?
   1. $V = \frac{kQ}{r}$
   2. $V = \frac{kQ}{b}$
   3. $V = 0$
   4. $V = \frac{kQ}{a}$