Inductance: Solid Cylindrical Conductor

To calculate the inductance of a solid cylindrical conductor, consider a 1-meter section of a non-magnetic, current-carrying conductor with radius r. Disregarding end effects and assuming uniform current density, Ampere's law helps determine the magnetic field inside the conductor. This law states that the magnetic field intensity H is concentric and constant within the conductor.

Given the uniform current distribution, the magnetic field H_x and flux density B_x inside the conductor are calculated. The differential flux per unit length inside the conductor is then computed, leading to the total internal flux linkages, which derive the internal inductance per unit length.

To determine the magnetic field outside the conductor, Ampere's law is applied to a contour enclosing the entire current. This helps calculate the magnetic flux density and the differential flux outside the conductor. Integrating these values between two external points provides the external flux linkages and the resulting external inductance per unit length.

The total inductance at an external point is the sum of the internal and external flux linkages. This comprehensive approach ensures an accurate determination of total inductance.

When considering an array of cylindrical conductors, each carrying a current such that the total current equals zero, the flux linkage is calculated by summing the contributions from each conductor. This ensures the accurate calculation of inductance for complex conductor configurations, which is essential for optimizing the performance and efficiency of electrical systems. This method combines internal and external inductance to provide a complete understanding of the inductance in practical applications.