Electric Potential Energy & Potential
Calculate electric potential $V = \frac{kQ}{r}$ and potential energy $U = qV$ for point charges and charge distributions. Visualize equipotential surfaces, explore the relationship $\vec{E} = -\nabla V$, and analyze how charges move from high to low potential.
Maxwell's Equations Overview
Explore the four Maxwell's equations that unify electricity and magnetism: Gauss's law, Gauss's law for magnetism, Faraday's law, and Ampère-Maxwell law. Visualize how these equations predict electromagnetic wave propagation and the interconnection between electric and magnetic fields.
Capacitor & Dielectric
Explore capacitance $C = \frac{Q}{V}$ and how dielectric materials increase capacitance by reducing the electric field. Visualize energy storage $U = \frac{1}{2}CV^2$ in parallel-plate capacitors and analyze the effects of dielectric constant on charge distribution.
Biot-Savart Law & Magnetic Field
Calculate magnetic fields using the Biot-Savart law $d\vec{B} = \frac{\mu_0}{4\pi}\frac{Id\vec{l} \times \hat{r}}{r^2}$ for current-carrying wires. Visualize how current elements contribute to the total magnetic field and explore field patterns around straight wires, loops, and complex geometries.
Electric Field & Gauss's Law
Apply Gauss's law $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ to calculate electric fields for symmetric charge distributions. Visualize electric flux through Gaussian surfaces and solve for fields around spheres, cylinders, and infinite planes using symmetry arguments.
Ampère's Law
Apply Ampère's law $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ to calculate magnetic fields in symmetric configurations. Visualize magnetic field patterns inside solenoids, toroids, and around current-carrying wires using closed loop integration.
RL Circuit Transient Response
Analyze RL circuit behavior with exponential current growth $I(t) = I_0(1 - e^{-t/\tau})$ where $\tau = L/R$ is the time constant. Visualize how inductors resist current changes, store magnetic energy, and create transient responses when switches open or close.
Faraday's Law & Lenz's Law
Apply Faraday's law $\mathcal{E} = -\frac{d\Phi_B}{dt}$ to calculate induced EMF from changing magnetic flux. Visualize how Lenz's law determines the direction of induced current to oppose flux changes, and explore applications in generators and transformers.