Relation between capacitance and input impedance:

Introduction:
The relationship between capacitance and input impedance is an important concept in the field of electrical engineering. Capacitance refers to the ability of a capacitor to store electrical energy, while input impedance refers to the resistance that a circuit presents to the input signal. Understanding the relationship between these two parameters is crucial in designing and analyzing electronic circuits.

Direct Proportionality:
The correct answer to the given question is option 'A', which states that the capacitance and input impedance are directly proportional to each other. This means that as the capacitance increases, the input impedance also increases, and vice versa.

Explanation:
1. Capacitance:
Capacitance is a fundamental property of a capacitor and is defined as the ratio of the charge stored on the capacitor to the voltage across it. It is denoted by the symbol 'C' and is measured in farads (F). The capacitance of a capacitor depends on factors such as the area of the capacitor plates, the distance between them (dielectric thickness), and the dielectric constant of the material between the plates.

2. Input Impedance:
Input impedance refers to the resistance that a circuit presents to the input signal. It is a complex quantity and is composed of a resistance component (real part) and a reactance component (imaginary part). The input impedance of a circuit depends on the components and configuration of the circuit. It is usually represented by the symbol 'Zin'.

3. Relationship:
The relationship between capacitance and input impedance arises due to the reactive nature of capacitors. Capacitors store energy in an electric field and exhibit reactance, which is the opposition to the flow of alternating current. Reactance is frequency-dependent and can be either capacitive or inductive.

When a capacitor is connected to a circuit, it introduces capacitive reactance, which is inversely proportional to the frequency of the input signal. As the capacitance increases, the capacitive reactance decreases, and as a result, the input impedance increases. This is because a larger value of capacitance allows more charge to be stored, thereby reducing the opposition to the flow of current.

Conclusion:
In summary, the capacitance and input impedance are directly proportional to each other. As the capacitance increases, the input impedance also increases. This relationship arises due to the reactive nature of capacitors and the introduction of capacitive reactance in the circuit. Understanding this relationship is crucial in designing electronic circuits and analyzing their performance.