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Chapter 3: Problem 41

The dimensions of formula of capacitance is a. \(\left[M^{-1} L^{-2} T^{\prime} A^{2}\right]\) b. \(\left[M^{-1} L^{-2} T^{3} A^{2}\right]\) c. \(\left[M^{-1} L^{-2} T^{4} A^{2}\right]\) d. \(\left[M^{-1} L^{-2} T^{2} A^{2}\right]\)

Short Answer

Expert verified
Option c is correct: \([M^{-1} L^{-2} T^{4} A^{2}]\).

Step by step solution

01

Understand the formula of capacitance

The formula for capacitance is \( C = \frac{Q}{V} \), where \( C \) is capacitance, \( Q \) is charge, and \( V \) is voltage. Understanding this is essential for deriving the unit and the dimensional formula.
02

Analyze the dimensions of charge (Q)

Charge \( Q \) has the dimensions \([I][T]\), where \([I]\) is the dimension of current (\(A\)), and \([T]\) represents time (\(T\)). Thus, the dimensional formula of \( Q \) is \([ A ][ T ]\).
03

Analyze the dimensions of voltage (V)

Voltage \( V \) can be derived from electric potential energy per unit charge: \( V = \frac{W}{Q} \), where \( W \) is work (or energy). The dimension of energy is \([M][L^2][T^{-2}]\), so the dimensional formula of voltage is \(\left[ M L^2 T^{-2} A^{-1} \right]\).
04

Derive the dimensional formula of capacitance

Plugging the derived dimensions into the capacitance formula \( C = \frac{Q}{V} \), we get \( C = \frac{[A][T]}{[M][L^2][T^{-2}][A^{-1}]} \). Simplifying this gives the dimensional formula \([M^{-1} L^{-2} T^4 A^2]\).
05

Match the derived dimensional formula with given options

Compare the derived dimensional formula \([M^{-1} L^{-2} T^4 A^2]\) with the given options. Option c matches exactly: \([M^{-1} L^{-2} T^4 A^2]\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Capacitance
Capacitance is a measure of a capacitor's ability to store electric charge. It's defined by the equation \( C = \frac{Q}{V} \), where \( C \) is the capacitance, \( Q \) is the electrical charge in coulombs, and \( V \) is the voltage or electric potential difference across the capacitor in volts.
This relationship implies that the greater the capacitance, the more charge a capacitor can store at a given voltage. Capacitance is typically measured in farads (\( F \)), a unit that reflects the ratio of charge to voltage.
  • Understanding Capacitance: In practice, capacitors can store significant amounts of energy even at low voltages if they have a large capacitance value.
  • Practical Applications: Capacitors are widely used in electronic circuits for a variety of applications such as smoothing output voltages in power supplies and in timing circuits.
Charge
Electric charge, denoted as \( Q \), is a fundamental property of matter that experiences a force in an electric field. It comes in two types: positive and negative.
  • Basic Properties of Charge: Like charges repel each other, while opposite charges attract. This is a basic principle of electromagnetism.
  • Units and Dimensions: Charge is measured in coulombs, with dimensions \([A][T]\), representing current (\(I\)) times time (\(T\)).
It's crucial to understand how charge interacts with electric fields, as this interaction is central to numerous phenomena in physics, including capacitance.
Voltage
Voltage, often called electric potential difference, is the measure of electric potential energy per unit charge. It's represented by the formula \( V = \frac{W}{Q} \), where \( W \) is work done or energy transferred, and \( Q \) is charge.
  • Units and Dimensions: Voltage is measured in volts (V), with the dimensional formula \([M L^2 T^{-2} A^{-1}]\).
Voltage tells us how much potential energy is available to move charges through an electric field or circuit. The higher the voltage, the greater the potential energy per unit charge.
Electric Potential Energy
Electric potential energy is the energy that a charge possesses due to its location in an electric field. It represents the work needed to move a charge against an electric field.
  • Relationship with Voltage: Since voltage is defined in terms of electric potential energy per charge, understanding this concept helps in analyzing voltage-related phenomena.
  • Dimensional Analysis: The dimension of energy is \([M L^2 T^{-2}]\), the same as work, emphasizing the interaction between force and displacement in an electric context.
In practical terms, electric potential energy is a key factor in determining how charges are distributed across capacitors and their subsequent effect on circuits.

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Most popular questions from this chapter

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