In Fig. 25-56, the parallel-plate capacitor of plate area 2.00 ×10^-2 m^2 is filled with two dic

step 1 Let capacitance 1 is filled with a material whose dialectic constant K1 is equal to 3, and C2 is

In Fig. 25-56, the parallel-plate capacitor of plate area...

In Fig. $25-56,$ the parallel-plate capacitor of plate area $2.00 \times 10^{-2} \mathrm{m}^{2}$ is filled with two diclectric slabs, each with thickness 2.00 $\mathrm{mm}$ . One slab has di- electric constant 3.00 , and the other, $4.00 .$ How much charge does the 7.00 $\mathrm{V}$ battery store on the capacitor?

In Fig. $25-56,$ the parallel-plate
capacitor of plate area $2.00 \times 10^{-2} \mathrm{m}^{2}$
is filled with two diclectric slabs, each with thickness 2.00 $\mathrm{mm}$ . One slab has di-
electric constant 3.00 , and the other,
$4.00 .$ How much charge does the 7.00 $\mathrm{V}$
battery store on the capacitor?

Key Concepts

Parallel-Plate Capacitor

A parallel-plate capacitor consists of two conductive plates separated by an insulating medium (or gap). Its ability to store electrical energy, characterized by the capacitance, depends on the area of the plates, the separation distance, and the properties of the material between the plates. This basic configuration is a fundamental model used to explore electric fields, energy storage, and the behavior of capacitors in circuits.

Dielectric Material

A dielectric is an insulating material introduced between the plates of a capacitor to increase its capacitance by reducing the effective electric field. The property of a dielectric that quantifies its effect is given by its dielectric constant (also known as relative permittivity). This constant indicates how much the material can enhance the capacitance compared to vacuum, and it is crucial in tailoring capacitor performance for various applications.

Capacitance Calculation

The capacitance of a capacitor is defined as the ratio of the stored charge to the applied voltage (C = Q/V). For a parallel-plate capacitor, the capacitance can be calculated by taking into account factors such as the area of the plates, the separation between them, and any dielectric material present. When dielectrics are involved, the capacitance is modified by the dielectric constant(s) of the material(s) used.

Series Combination of Dielectric Layers

When a capacitor is filled with more than one dielectric material arranged in layers along the direction of the electric field, the system behaves like a series combination of capacitors. In this case, the effective capacitance is determined by taking the sum of the reciprocals of the individual capacitances corresponding to each dielectric layer. This concept is essential when multiple materials are used to fill the gap of a capacitor, as it influences the overall charge storage capability.

Relationship Between Voltage and Stored Charge

The stored charge on a capacitor is directly determined by the product of its effective capacitance and the voltage applied across it (Q = CV). Understanding this relationship is key in circuit analysis and energy storage applications, as it links the physical properties of the capacitor (modified by any dielectric materials) to its operational performance when connected to a voltage source such as a battery.

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