Question

State, explain and compare features of
(i) Gauss’s law in electrostatics and
(ii) Ampere’s circuital law in magnetostatics.

Hint:

Gauss → surfaces and charge. Ampere → loops and current.

Answer 1

Step 1: Gauss’s law in electrostatics.
Statement: The total electric flux through any closed surface is equal to $\dfrac{1}{\varepsilon_0}$ times the total charge enclosed. \[ \oint \vec{E} \cdot d\vec{A} = \frac{q_{enclosed}}{\varepsilon_0} \] - Useful for highly symmetric charge distributions (spherical, cylindrical, planar). - Provides relation between electric field and enclosed charge.
Step 2: Ampere’s circuital law in magnetostatics.
Statement: The line integral of magnetic field $\vec{B}$ around a closed path is equal to $\mu_0$ times the net current enclosed by the path. \[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed} \] - Useful for symmetric current distributions (solenoids, toroids, straight conductors). - Provides relation between magnetic field and enclosed current.
Step 3: Comparison.
- Gauss’s law $\leftrightarrow$ electric field and charge (flux law).
- Ampere’s law $\leftrightarrow$ magnetic field and current (circulation law).
- Both are integral laws, derived from Maxwell’s equations. - Gauss’s law uses closed surfaces, Ampere’s law uses closed loops.
Step 4: Conclusion.
Both laws are symmetry tools: Gauss’s law for $\vec{E}$, Ampere’s law for $\vec{B}$.