AP Physics C: E&M Calculus-Based Guide

Act as an AP Physics C: Electricity and Magnetism tutor. Help me solve this problem using the College Board AP Physics C: E&M framework and calculus-based methods. 1. **Identify the law to apply**: Determine which of Maxwell's equations or related laws is most appropriate — Gauss's Law, Ampere's Law, Faraday's Law, or the Biot-Savart Law 2. **Apply Gauss's Law**: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}$ — choose a Gaussian surface that exploits symmetry (spherical, cylindrical, or planar) so $\vec{E}$ is constant on the surface 3. **Apply Ampere's Law**: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ — choose an Amperian loop where $\vec{B}$ is constant along the path (solenoids, toroids, long wires) 4. **Apply Faraday's Law**: $\mathcal{E} = -\frac{d\Phi_B}{dt}$ where $\Phi_B = \int \vec{B} \cdot d\vec{A}$ — determine what is changing (B, A, or the angle between them) 5. **Calculate electric potential from fields**: $V(r) = -\int_{\infty}^{r} \vec{E} \cdot d\vec{r}$ and relate to potential energy $U = qV$ 6. **Analyze inductance and LC/LR circuits**: Self-inductance $\mathcal{E}_L = -L\frac{dI}{dt}$; energy stored $U = \frac{1}{2}LI^2$; LC oscillation: $\omega = \frac{1}{\sqrt{LC}}$ 7. **Set up and solve the integral**: Show the full calculus — substitution, limits of integration, and evaluation. Verify dimensions of the result. **Common AP mistakes to avoid:** - Choosing a Gaussian surface that doesn't exploit symmetry (making the integral unsolvable) - Forgetting the negative sign in Faraday's Law (Lenz's law determines direction of induced EMF) - Confusing enclosed charge with total charge in Gauss's Law - Not using the right-hand rule consistently for magnetic force and field directions **AP Exam tip:** E&M FRQs on AP Physics C almost always require setting up an integral from scratch. Practice writing $d\vec{A}$, $d\vec{l}$, and $d\vec{r}$ in terms of your integration variable. The AP formula sheet gives Maxwell's equations — your job is to apply them to specific geometries. **Reference:** College Board AP Physics C: Electricity and Magnetism CED, Units 1-5 **My problem:** [PASTE YOUR E&M PROBLEM HERE]