AP Physics C: E&M - Advanced Concepts & Calculations

Front

Derive the electric field magnitude (E) inside a solid insulating sphere of uniform volume charge density (rho) at a distance r < R.

Back

Use Gauss's Law. Choose a spherical Gaussian surface of radius r. The enclosed charge is Q_enc = rho * (4/3) * pi * r^3. Flux is Phi = E * 4 * pi * r^2. Solving for E: E * (4 * pi * r^2) = (rho * (4/3) * pi * r^3) / epsilon_0. Result: E = (rho * r) / (3 * epsilon_0). The field increases linearly with distance r from the center.

Front

Describe the charge distribution and electric field inside a conductor at electrostatic equilibrium.

Back

1. Excess charge resides entirely on the outer surface. 2. The electric field inside the conducting material is zero (E=0). 3. The entire conductor is an equipotential volume. 4. The electric field just outside the surface is perpendicular to the surface with magnitude sigma / epsilon_0.

Front

Explain the behavior of a spherical conducting shell with a point charge +q placed at its center.

Back

To maintain E=0 inside the conductor material, an inner surface charge of -q is induced (residing on the inner wall). Conservation of charge requires the outer surface to acquire a net charge of +q (in addition to any pre-existing charge). If the shell is grounded, the outer surface charge drains away, leaving the shell neutral but shielding the exterior from the internal +q.

Front

Derive the capacitance of a cylindrical capacitor (two concentric cylinders of length L, radii a and b).

Back

Gauss's Law on a cylinder of radius r (a < r < b) gives E = lambda / (2 * pi * epsilon_0 * r), where lambda is charge per unit length. Integrate E from a to b: delta V = -integral_a^b (lambda / (2 * pi * epsilon_0 * r)) dr = (lambda / (2 * pi * epsilon_0)) * ln(b/a). Capacitance per length is C/L = Q / (L * delta V) = lambda / delta V = 2 * pi * epsilon_0 / ln(b/a).

Front

Calculate the equivalent capacitance of two capacitors in series versus parallel if C1 = 2 * C2.

Back

Let C2 = C. C1 = 2C. Parallel: C_eq = C1 + C2 = 3C. Series: 1 / C_eq = 1/C1 + 1/C2 = 1/(2C) + 1/C = 3 / (2C). Therefore, C_eq = (2/3) * C. Series reduces capacitance; parallel increases it.