Gauss's Law - AP Physics C: Electricity and Magnetism

Spherical surface. Spherical symmetry matches point charge field distribution.

$\nabla \bullet \textbf{E} = \frac{\rho}{\text{ε}_0}$. Local form showing field divergence at each point in space.

Planar symmetry. Infinite plane creates uniform field perpendicular to surface.

$\rho$. Volume charge density measured in coulombs per cubic meter.

Requires symmetry for simplification. Without symmetry, integral cannot be easily evaluated.

$\Phi_E = \frac{Q_{\text{enc}}}{\epsilon_0}$. Relates electric flux through closed surface to enclosed charge.

Integral form. Allows calculation of electric field from enclosed charge.

Planar symmetry. Infinite plane creates uniform field perpendicular to surface.

$\text{ε}_0$. Permittivity constant relating flux to enclosed charge.

$E = \frac{Q}{4\text{π}\text{ε}_0 r^2}$. Spherical symmetry gives $1/r^2$ dependence outside sphere.

Requires symmetry for simplification. Without symmetry, integral cannot be easily evaluated.

$E = \frac{\text{λ}}{2\text{π}\text{ε}_0 r}$. Cylindrical symmetry gives $1/r$ dependence for line charge.

Electric flux is proportional to enclosed charge. Gauss's Law shows flux equals enclosed charge divided by $\text{ε}_0$.

They are perpendicular to the surface. Electric field must be normal to conducting surface.

$\rho$. Volume charge density measured in coulombs per cubic meter.

Permittivity of free space. Fundamental constant equal to $8.85 \times 10^{-12}$ F/m.

Divergence of the electric field. Measures how much electric field spreads out from a point.

Both describe electric fields from charge distributions. Both express how charges create electric fields in space.

$E = \frac{\rho r}{3\text{ε}_0}$. Field increases linearly with radius inside uniform sphere.

$E = \frac{\text{σ}}{2\text{ε}_0}$. Field from infinite plane with surface charge density $\text{σ}$.

Zero. No charge enclosed means zero field by Gauss's Law.

Symmetrical Gaussian surface. Surface shape matching charge distribution symmetry simplifies calculations.

Simplifies electric field calculation. Allows electric field to be factored from surface integral.

Gauss's Law. Fundamental relationship between field sources and field behavior.

Field lines entering equal to those leaving if no charge inside. Net flux through closed surface equals enclosed charge.

A Gaussian surface. Imaginary closed surface chosen to exploit charge distribution symmetry.

Total charge enclosed by the surface. Sum of all charges within the Gaussian surface boundary.

Zero. No enclosed charge means equal field lines enter and exit.

Spherical symmetry. Charge distribution has same properties in all radial directions.