Ampère's law Knowledge-Based Study

Study notes

1. Fundamental Concept

   Ampère's circuital law relates the integrated magnetic field around a closed loop to the electric current passing through the surface bounded by that loop. It serves as a magnetic analogue to Gauss's law in electrostatics, offering an elegant method to calculate magnetic fields for configurations possessing high degrees of spatial symmetry.

   Mathematical Formulation

   In magnetostatics, for steady currents, the law is mathematically stated as:

   $$\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\text{enc}}$$

   where:

   • $\vec{B}$ is the magnetic field vector at each point along the integration path.
   • $d\vec{\ell}$ is an infinitesimal vector displacement directed tangent to the path.
   • $\mu_0$ is the permeability of free space, defined as $4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$.
   • $I_{\text{enc}}$ is the net macroscopic electric current piercing the open surface bounded by the closed loop.

   Sign Convention and the Right-Hand Rule

   The algebraic sign of the enclosed current is determined by the direction chosen for the line integral. If you curl the fingers of your right hand in the direction of traversal along the loop, your thumb points in the direction of positive current.

   Choosing an Amperian Loop

   While Ampère's law is universally true for steady currents, it is only practically useful for calculating the magnetic field when symmetry allows $\vec{B}$ to be factored out of the integral.

   Criteria for Useful Amperian Loops

   An Amperian loop is an imaginary, closed geometric path chosen such that along each segment, one of the following conditions is satisfied:

   • Tangential and Uniform: The field $\vec{B}$ is parallel to $d\vec{\ell}$ and its magnitude is constant, simplifying the dot product to $B \oint d\ell = B L$.
   • Perpendicular: The field $\vec{B}$ is perpendicular to $d\vec{\ell}$, resulting in $\vec{B} \cdot d\vec{\ell} = 0$.
   • Zero Field: The magnetic field $\vec{B}$ is known to be zero along that portion of the path.

   The standard geometries that permit this simplification are cylindrical symmetry (long wires, coaxial cables), planar symmetry (infinite conducting sheets), and toroidal/helical symmetry (solenoids and toroids).

   Field Outside and Inside a Uniform Wire

   Consider an infinitely long, straight solid cylindrical wire of radius $R$ carrying a uniformly distributed steady current $I$.

   Region Outside the Conductor ($r \ge R$)

   Choose a circular Amperian loop of radius $r$ concentric with the wire. By cylindrical symmetry, $\vec{B}$ is everywhere tangent to the circle and uniform in magnitude:

   $$\oint \vec{B} \cdot d\vec{\ell} = B(2\pi r) = \mu_0 I_{\text{enc}}$$

   Because the loop encloses the entire current, $I_{\text{enc}} = I$. Solving for $B$ gives:

   $$B = \frac{\mu_0 I}{2\pi r}$$

   Region Inside the Conductor ($r < R$)

   For an internal circular loop of radius $r < R$, only a fraction of the current is enclosed. Assuming a uniform current density $J = \frac{I}{\pi R^2}$:

   $$I_{\text{enc}} = J (\pi r^2) = I \left(\frac{r^2}{R^2}\right)$$

   Applying Ampère's law:

   $$B(2\pi r) = \mu_0 I \frac{r^2}{R^2} \implies B = \frac{\mu_0 I r}{2\pi R^2}$$

   Thus, the magnetic field increases linearly with radial distance from the center, reaches its maximum at the surface $r = R$, and decays inversely with distance outside the wire.

   Ideal Solenoids

   An ideal solenoid consists of a closely wound helical coil whose length is much greater than its radius. Inside the solenoid, the magnetic field is uniform and directed parallel to the central axis; outside, the field approaches zero.

   To evaluate the field, consider a rectangular Amperian loop of length $L$ with one side inside the solenoid, parallel to the axis, and the opposite side outside:

   1. Inside segment: $\int \vec{B} \cdot d\vec{\ell} = B L$
   2. Two perpendicular segments: $\vec{B} \cdot d\vec{\ell} = 0$ because $\vec{B}$ is perpendicular to the path or negligible.
   3. Outside segment: $\vec{B} \approx 0$, giving no contribution.

   If the solenoid has $n$ turns per unit length, the enclosed current is $I_{\text{enc}} = n L I$. Applying Ampère's law yields:

   $$B L = \mu_0 (n L I) \implies B = \mu_0 n I$$

   Toroids

   A toroid is a solenoid bent into a continuous circle with $N$ total turns. Using a circular Amperian loop of radius $r$ inside the coil winding:

   $$B(2\pi r) = \mu_0 N I \implies B = \frac{\mu_0 N I}{2\pi r}$$

   The magnetic field is strictly confined within the core of the toroid and is zero in the central opening and outside the ring.

   Scope of Validity

   Ampère's law in the form $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\text{enc}}$ is strictly valid only under magnetostatic conditions, where electric fields are constant in time and currents are steady and continuous.

   The Continuity Contradiction

   When considering a time-varying electric system, such as a charging capacitor, applying Ampère's law to different surfaces bounded by the same loop yields conflicting results: a flat surface passing between the plates has $I_{\text{enc}} = 0$, whereas a bulging surface cutting the wire has $I_{\text{enc}} = I$.

   Displacement Current

   James Clerk Maxwell resolved this contradiction by introducing the displacement current, defined as $I_d = \varepsilon_0 \frac{d\Phi_E}{dt}$, where $\Phi_E$ is the electric flux. The complete Maxwell-Ampère law states:

   $$\oint \vec{B} \cdot d\vec{\ell} = \mu_0 \left( I_{\text{enc}} + \varepsilon_0 \frac{d\Phi_E}{dt} \right)$$

   This generalized equation shows that both moving charges (conduction current) and changing electric fields produce magnetic fields, ensuring universal conservation of charge and laying the foundation for electromagnetic wave theory.