Question

A series R−C circuit is connected to AC voltage source. Consider two cases; (A) when C is without a dielectric medium and (B) when C is filled with a dielectric of constant 4. The current I ​through the resistor and voltage across the capacitor are compared in the two cases. Which of the following is/are true?
A.$\begin{array}{rl}& I_R^A>I_R^B\end{array}$
B.$\begin{array}{rl}& I_R^A<I_R^B\end{array}$
C.$\begin{array}{rl}& V_C^A>V_C^B\end{array}$
D.$\begin{array}{rl}& V_C^A<V_C^B\end{array}$

Answer 1

Hint: Here, we first find the relation between the current flowing through the resistor and voltage across the capacitor in a series R-C circuit. Further, by observing the two given cases we can deduce a final expression which will give as the required result to this question.
Formula used:
$\begin{array}{rl}& I={\displaystyle \frac{V}{Z}}\\ & V^2=(V_R{)}^2+({V_C}^2)\end{array}$

Complete answer:
In the given two cases where first is when the capacitor is without a dielectric medium and second is when the capacitor is filled with dielectric constant. Here, when the current I through the resistor and voltage across the capacitor are compared with each other, we get the relation as:
$\begin{array}{rl}& I={\displaystyle \frac{V}{Z}}\\ & \because V^2=(V_R{)}^2+({V_C}^2)\end{array}$
$\begin{array}{rl}& \Rightarrow V^2=(IR{)}^2+\left({\displaystyle \frac{I}{\omega C^2}}\right)\\ & \therefore Z^2=R^2+\left({\displaystyle \frac{1}{\omega C^2}}\right)\end{array}$

Therefore, from this result we can say that option B) and option C) both are correct.

Additional information:
Here, as we know that a capacitor is defined as a device that stores electrical energy in an electric field. Capacitor is a passive electronic component with two terminals. The effect of a capacitor is called capacitance.
Further, if we go in detail of a capacitor, a capacitor consists of two or more parallel conductive or metal plates. These plates are not connected or touching each other, but rather they are electrically separated either by air or by some form of a good insulating material like waxed paper, mica, ceramic, plastic or some form of a liquid gel as used in electrolytic capacitors. This insulating layer between capacitor’s plates is commonly called the Dielectric.
The dielectric constant is defined as a measure of the amount of electric potential energy, in the form of induced polarization which is stored in a given volume of material under the action of an electric field.

Note:
Capacitance is measured in Farad, or can be said as one coulomb per volt. Capacitance is dependent on the dielectric constant as well as on the distance between the two plates. The parallel plate capacitor is the simplest form of capacitor.