A dielectric of permittivity 3.5 ×10^-11 F/m completely fills the volume between two capacitor p

step 1 Okay, so we're doing chapter 29, problem 65. So we have two capacitor plates with a dielectric i

A dielectric of permittivity 3.5 ×10^-11 F/m completely...

A dielectric of permittivity 3.5 $\times$ 10$^{-11}$ F/m completely fills the volume between two capacitor plates. For t 7 0 the electric flux through the dielectric is (8.0 $\times$ 10$^3$ V $\cdot$ m$/s^3)t^3$. The dielectric is ideal and nonmagnetic; the conduction current in the dielectric is zero. At what time does the displacement current in the dielectric equal 21 $\mu$A ?

Key Concepts

Maxwell's Displacement Current

This concept arises from Maxwell's amendment to Ampère’s law, which states that a time-varying electric field produces a current-like effect called the displacement current. In regions where there is no conduction current, like inside the dielectric of a capacitor, the changing electric field generates a displacement current that contributes to the magnetic field. This current is mathematically equivalent to the time derivative of the electric displacement field integrated over a surface.

Dielectric Permittivity

Dielectric permittivity is a material property that quantifies how an electric field affects, and is affected by, a dielectric medium. It relates the electric field E to the electric displacement D by the relation D = ?E. In problems involving capacitors, permittivity determines the degree to which the dielectric is polarized by the field, thereby influencing both the storage of electrical energy and the generation of displacement current when the electric field is time-dependent.

Time-Varying Electric Fields and Their Derivatives

When electric fields vary with time, their flux (or the related displacement field) is also time-dependent, and its rate of change (derivative with respect to time) is central to determining the displacement current. The analysis of such systems involves differentiating the time-dependent functions that describe the electric flux or displacement field to find how rapidly they change, which is directly proportional to the displacement current according to Maxwell’s equations.

Capacitors with Dielectrics

Capacitors store electrical energy by maintaining an electric field between two conductors, and the inclusion of a dielectric material enhances this storage capability. The dielectric modifies the electric field between the plates by becoming polarized, which affects both the capacitance and the dynamic behavior of the system when the applied voltage is time-dependent. Understanding how dielectrics behave under varying fields is essential for analyzing displacement currents in such configurations.

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