SUMMARY
The maximum potential in a coaxial cable configuration is achieved when the inner conductor radius \( a \) is set to \( b/e \), where \( b \) is the outer conductor radius and \( e \) is the base of the natural logarithm. Utilizing Gauss's Law, the electric field \( E \) for the region between the conductors is expressed as \( E = \frac{2k\lambda}{r} \), with \( \lambda \) representing the charge per unit length \( Q/L \). The potential difference \( V \) is derived from the integral of the electric field, resulting in \( V = 2k\lambda \ln(b/a) \). To find the maximum potential, one must differentiate \( V \) with respect to \( a \) and set the derivative to zero.
PREREQUISITES
- Understanding of Gauss's Law in electrostatics
- Familiarity with electric field and potential difference concepts
- Knowledge of logarithmic functions and their properties
- Basic calculus, specifically differentiation techniques
NEXT STEPS
- Study the application of Gauss's Law in cylindrical coordinates
- Learn about electric field calculations in coaxial cable systems
- Explore differentiation techniques for optimizing functions
- Investigate the physical implications of potential differences in electrical engineering
USEFUL FOR
Students and professionals in electrical engineering, physicists studying electromagnetism, and anyone interested in optimizing coaxial cable designs for maximum potential.