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

By choosing an appropriate path of integration. Symmetry allows field magnitude to be constant along path.

It encloses the current for calculating the magnetic field. The path defines which currents contribute to the magnetic field.

The current enclosed by the path of integration. Only current passing through the integration loop contributes to the field.

The curl of the magnetic field $\textbf{B}$. Mathematical operation measuring circulation of magnetic field.

$B = \frac{\mu_0 J r}{2}$ for $r < R$. Field proportional to current density and distance from center.

$\oint \mathbf{B} \cdot d\mathbf{l} = 0$ when no current is enclosed. No enclosed current means zero circulation around the loop.

$B = \text{μ}_0 n I$. Field strength depends on turn density and current.

The displacement current term, $\text{μ}_0 \text{ε}_0 \frac{\text{∂}\textbf{E}}{\text{∂}t}$. Accounts for changing electric fields creating magnetic effects.

Circular and perpendicular to the wire. Follows right-hand rule with current direction.

Number of turns per unit length of the solenoid. Turn density determines field strength in solenoid.

$B = \frac{\text{μ}_0 I r}{2 \text{π} R^2}$ for $r < R$. Field grows linearly with radius inside the conductor.

The direction of the magnetic field. Thumb points along current, fingers curl with field direction.

The curl of the magnetic field $\textbf{B}$. Mathematical operation measuring circulation of magnetic field.

Computing the magnetic field around a closed loop. Relates field circulation to total current through the loop.

The magnetic field must be steady and symmetric. Required for the mathematical simplification of the integral.

Tesla (T). Standard SI unit for magnetic flux density.

$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$. Includes displacement current for time-varying electric fields.

$\oint \textbf{B} \bullet \text{d}\textbf{l} = \mu_0 I_{\text{enc}}$. The line integral of magnetic field equals permeability times enclosed current.

Both relate a field to its source within a closed surface or loop. Both use closed surfaces/loops to relate fields to sources.

A term accounting for changing electric fields. Maxwell's correction for time-varying electromagnetic fields.

Simplifies the calculation of the magnetic field. High symmetry allows field magnitude to factor out of integral.

Zero, assuming an ideal solenoid. Field lines are confined within the solenoid.

The field decreases as $\frac{1}{r}$. Inverse relationship due to field spreading over larger circumference.

To find the field between the inner and outer conductors. Field exists only in regions with net enclosed current.

Around a long straight current-carrying wire. High symmetry makes Ampère's Law calculations straightforward.

The current enclosed by the path of integration. Only current passing through the integration loop contributes to the field.

Both describe magnetic fields due to currents. Both are fundamental laws for calculating magnetic fields.

$4\text{π} \times 10^{-7} \text{ T m/A}$. Fundamental constant relating magnetic field to current.

The permittivity of free space. Fundamental constant relating electric field to charge.