Problem 37 - Master Resource Book in JEE Main Physics

Question

Farad is not equivalent to (a) \(\frac{q}{v}\) (b) \(q V^{2}\) (c) \(\frac{q^{2}}{J}\) (d) \(\frac{J}{v^{2}}\) \((q=\) coulomb, \(V=\) volt and \(J=\) joule \()\)

Step-by-Step Solution

Answer

Option (b) \(q V^{2}\) is not equivalent to a farad.

Step 1: Define Farad

Farad is the SI unit of capacitance. It is defined as the ability of a system to store a charge per unit voltage. Mathematically, it is given by \( C = \frac{Q}{V} \), where \( C \) is the capacitance in farads, \( Q \) is the charge in coulombs, and \( V \) is the voltage in volts.

Step 2: Analyze Option (a)

Option (a) is \( \frac{q}{v} \). In the context of capacitance, this formula represents farads, as it is equivalent to \( C = \frac{Q}{V} \). Thus, option (a) is equivalent to a farad.

Step 3: Analyze Option (b)

Option (b) is \( q V^{2} \). Here, \( q \) is charge in coulombs and \( V \) is voltage in volts. This expression represents a physical quantity with the dimension of \( C \times (V^2) \), which does not correspond to the unit of capacitance (farad). Therefore, option (b) is not equivalent to a farad.

Step 4: Analyze Option (c)

Option (c) is \( \frac{q^{2}}{J} \). By analyzing the energy stored in a capacitor, we find \( \text{Energy, } E = \frac{1}{2} C V^2 = \frac{1}{2} \frac{Q^2}{C} \). Rearranging gives \( C = \frac{Q^2}{2E} \). This closely relates to the structure of \( \frac{q^2}{J} \) showing \( \frac{q^2}{J} \) is akin to \( C \), or capacitance in farads. Hence, option (c) is equivalent to a farad.

Step 5: Analyze Option (d)

Option (d) is \( \frac{J}{v^{2}} \). Using energy equivalence \( E = \frac{1}{2} CV^2 \), rearranging gives \( C = \frac{2J}{V^2} \). Therefore, \( \frac{J}{V^2} \) is functionally related to capacitance in farads when scaled, and thus option (d) is equivalent to a farad.

Key Concepts

SI unit of capacitance

The SI unit of capacitance is the farad. But what is capacitance all about? Capacitance is essentially a measure of how much electric charge, measured in coulombs, a system can store per unit of electric potential difference, measured in volts. The official symbol for capacitance is \( C \). It's a crucial concept in electrical circuits, especially when dealing with capacitors, which are devices designed to store and release electrical energy.
For something to be measured in farads, it means that if one coulomb of charge increases the potential difference by one volt, the capacitance is exactly one farad. This unit plays a significant role in designing and understanding circuits and electronic devices.
The beauty of the SI units is their universality and consistency, providing a standardized way to express and understand physical quantities across varying fields.

farad

The farad is the fundamental SI unit for measuring capacitance. Named after Michael Faraday, a renowned scientist, the farad quantifies how effectively an electrical component can store charge.

• A large capacitance (more farads) indicates a strong ability to store more charge at a given voltage.
• One farad is a rather large unit, and practical capacitors often measure in microfarads (\( \mu F = 10^{-6} \text{ F} \)), nanofarads (\( nF = 10^{-9} \text{ F} \)), or picofarads (\( pF = 10^{-12} \text{ F} \)).

Understanding the use of farads in circuits is essential, as it affects how the circuit stores and uses energy.
Think of the farad in terms of its real-world application: capacitors hold charge to smooth out electric signals, providing stability and efficiency in devices ranging from camera flashes to computer processors.

charge-voltage relationship

The charge-voltage relationship is foundational to understanding how capacitors work. It's expressed by the formula \( C = \frac{Q}{V} \), where \( C \) is capacitance, \( Q \) is charge, and \( V \) is voltage. This relationship tells us that capacitance is the ratio of charge stored to the potential difference across it.
This is essential because:

• A higher capacitance means more charge can be stored at a given voltage.
• It allows for quantifying how changes in voltage affect the stored charge.

By understanding this relationship, one can predict how a capacitor will behave in different electronic configurations. It’s significant in designing efficient circuits that can maintain desired voltages and manage power effectively.