The voltage across a cylindrical conductor of radius r,...

The voltage across a cylindrical conductor of radius $r$, length $L$, and resistance $R$ varies with time. The time-varying voltage causes a time-varying current, $i$, to flow in the cylinder. Show that the displacement current equals $\epsilon_{0}$ odildt, where $\rho$ is the resistivity of the conductor.

Key Concepts

Continuity Equation in Electromagnetism

The continuity equation is a mathematical statement of the conservation of electric charge. It connects the rate of change of charge density to the divergence of current density. In scenarios with time-varying conduction currents, the displacement current must be included to satisfy the continuity equation, ensuring that all changes in charge are properly accounted for via either conduction or displacement effects.

Time-Varying Electric Fields

Time-varying electric fields arise when the applied voltage or charge distribution changes with time. In a resistive material like a cylindrical conductor, a changing voltage induces a changing electric field, which in turn gives rise to a displacement current. The dynamics of these fields are critical for analyzing circuits and electromagnetic phenomena where steady-state approximations no longer hold.

Ohm’s Law and Conduction Current

Ohm’s law relates the conduction current flowing through a material to the applied voltage and the material’s resistance. In a conductor, the current is determined by the local electric field and the material’s resistivity (or its reciprocal, conductivity). When the applied voltage varies with time, the conduction current also changes, thereby leading to time-dependent effects that must be taken into account alongside the displacement current.

Maxwell's Displacement Current

Maxwell’s displacement current is a term added to Ampere’s law to account for the effects of a time?varying electric field in situations where there is no conduction current. It is given by ?? times the time derivative of the electric field (???E/?t) and is essential in ensuring that the modified Ampere’s law is consistent with the conservation of charge. This concept is central to understanding the propagation of electromagnetic waves and the behavior of circuits with time-dependent voltages.

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