Internal and External Magnetic Field Distribution of a Current-Carrying Cylindrical Conductor

In the study of electromagnetism, magnetostatics serves as a fundamental pillar, focusing on the magnetic fields generated by steady, time-invariant currents. For engineers and physicists, the ability to accurately model the magnetic field distribution of specific geometric structures is not merely a theoretical exercise but a prerequisite for designing efficient electrical systems.

One of the most essential models in this field is the infinitely long, straight cylindrical conductor. This model provides a clear window into the behavior of magnetic fields and serves as a primary application for Ampère’s Circuital Law. By analyzing a conductor with radius $R$ carrying a total current $I$ distributed uniformly across its cross-section, we can observe how the magnetic field transitions from a linear growth within the material to an inverse decay outside of it.

Mathematical Framework and Symmetry

To solve for the magnetic field $\mathbf{B}$, we must first establish the physical parameters and the coordinate system. Given the cylindrical geometry, the cylindrical coordinate system $(r, \varphi, z)$ is the most efficient mathematical choice.

We assume the following:

• The conductor is infinitely long along the $z$-axis.
• The total current $I$ flows in the $z$-direction.
• The current density $J$ is uniform throughout the cross-section, defined as $J = \frac{I}{\pi R^2}$.

Due to the axial symmetry of the system, the magnetic field $\mathbf{B}$ must be independent of the angular position $\varphi$ and the longitudinal position $z$. Furthermore, the field lines must form concentric circles around the axis, meaning the direction of $\mathbf{B}$ is purely azimuthal (in the $\varphi$ direction). Consequently, the magnitude of the field, $B$, depends solely on the radial distance $r$ from the central axis.

The primary tool for this derivation is Ampère’s Circuital Law, which relates the integrated magnetic field around a closed loop to the net current passing through that loop:

$$\oint_L \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}$$

By selecting an Ampèrean loop as a circle of radius $r$ centered on the axis, the line integral simplifies to $B(2\pi r)$.

Magnetic Field Distribution Inside the Conductor ($r < R$)

When we investigate the region inside the conductor, the Ampèrean loop only encloses a fraction of the total current.

1. Calculating the Enclosed Current: Since the current is distributed uniformly, the current enclosed by a loop of radius $r$ is proportional to the ratio of the areas:
   $$I_{\text{enc}} = J \cdot (\pi r^2) = \left( \frac{I}{\pi R^2} \right) \pi r^2 = I \frac{r^2}{R^2}$$

2. Applying Ampère’s Law: Substituting the enclosed current into the integral form:
   $$B(2\pi r) = \mu_0 \left( I \frac{r^2}{R^2} \right)$$

3. Solving for $B$:
   $$B_{\text{internal}}(r) = \frac{\mu_0 I r}{2\pi R^2}$$

Key Observation: Inside the conductor, the magnetic field strength increases linearly with the distance $r$ from the axis. At the exact center ($r = 0$), the magnetic field is zero, reaching its peak value at the conductor's surface ($r = R$).

Magnetic Field Distribution Outside the Conductor ($r > R$)

For any point located outside the conductor, the Ampèrean loop encompasses the entire current flowing through the cylinder.

1. Determining the Enclosed Current: In this region, the net current enclosed is simply the total current:
   $$I_{\text{enc}} = I$$

2. Applying Ampère’s Law:
   $$B(2\pi r) = \mu_0 I$$

3. Solving for $B$:
   $$B_{\text{external}}(r) = \frac{\mu_0 I}{2\pi r}$$

Key Observation: Outside the conductor, the magnetic field behaves exactly as if the entire current were concentrated along a single, infinitely thin wire at the center. The field strength follows an inverse relationship with the distance $r$, meaning it decays as one moves further away from the conductor.

Comparative Analysis and Physical Continuity

By examining the two regions together, we can derive a complete profile of the magnetic field distribution. The transition at the boundary $r = R$ is a critical point of interest:

• Internal Profile ($r \le R$): $B \propto r$ (Linear growth).
• External Profile ($r \ge R$): $B \propto \frac{1}{r}$ (Hyperbolic decay).
• Boundary Continuity: At the surface $r = R$, both formulas yield the same value:
  $$B(R) = \frac{\mu_0 I}{2\pi R}$$
  This confirms that the magnetic field is continuous at the interface between the conductor and the surrounding space.

Engineering Significance

Understanding these distributions is vital for several practical applications in electrical engineering:

• Inductance Calculation: The internal magnetic field contributes significantly to the internal inductance of a conductor, a crucial factor in power system analysis.
• Electromagnetic Interference (EMI): In high-current applications, the external field can induce unwanted currents in nearby electronic components. Engineers use these models to design effective electromagnetic shielding.
• Coaxial Cable Design: The principle of managing internal and external fields is the foundation for designing coaxial cables, where the goal is to confine the electromagnetic field within the dielectric space to minimize radiation and interference.
• Eddy Current Analysis: In AC applications, the varying magnetic field within the conductor induces eddy currents, which lead to energy losses (skin effect). Detailed field mapping is necessary to mitigate these losses in high-frequency design.