Problem 114
Question
Match the following Column I with Column II. Column 1\. Electrical capacity II. Permittivity of free space III. Electrical potential IV. Electrical energy Column II (A) \(\left[\mathrm{M}^{-} \mathrm{L}^{-3} \mathrm{~T}^{4} \mathrm{~A}^{2}\right]\) (B) \(\left[\mathrm{M}^{\prime} \mathrm{L}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-1}\right]\) (C) \(\left[\mathrm{M}^{\prime} \mathrm{L}^{2} \mathrm{~T}^{-2}\right]\) (D) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^{4} \mathrm{~A}^{2}\right]\) (a) \(1-\mathrm{A}, \mathrm{ll}-\mathrm{D}, \mathrm{IIl}-\mathrm{B}, \mathrm{IV}-\mathrm{C}\) (b) \(1-D, 11-A, 111-C\), IV-C (c) \(1-\mathrm{D}, 11-\mathrm{A}, 1 \mathrm{ll}-\mathrm{C}, \mathrm{IV}-\mathrm{B}\) (d) \(1-A, 11-D, 11]-C, T V-B\)
Step-by-Step Solution
Verified Answer
The correct match is option (a): 1-D, II-D, III-B, IV-C.
1Step 1: Understand Column I Quantities
Identify the physical quantities given in Column I: Electrical capacity (capacitance), permittivity of free space, electrical potential (voltage), and electrical energy. Each of these has standard units in the SI system and corresponding dimensional formulas.
2Step 2: Identify Dimensional Formulas
Recall the dimensional formulas for each of the Column I quantities:1. Electrical capacity (capacitance) is \(\left[\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{~T}^4 \mathrm{~A}^2\right]\).2. Permittivity of free space is \(\left[\mathrm{M}^{-1} \mathrm{L}^{-3} \mathrm{~T}^4 \mathrm{~A}^2\right]\).3. Electrical potential (voltage) is \(\left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-1}\right]\).4. Electrical energy is \(\left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{~T}^{-2}\right]\).
3Step 3: Match Column I with Column II
Compare the dimensional formulas of Column I quantities with those presented in Column II:1. Electrical capacity (capacitance) can be matched incorrectly since none represents its correct formula.2. Permittivity of free space matches with D: \(\left[\mathrm{M}^{-1} \mathrm{L}^{-3} \mathrm{~T}^4 \mathrm{~A}^2\right]\).3. Electrical potential (voltage) matches with B: \(\left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-1}\right]\).4. Electrical energy matches with C: \(\left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{~T}^{-2}\right]\).
4Step 4: Verify Against Answer Options
Compare your matches with the provided options:a. (1-A, II-D, III-B, IV-C),b. (1-D, II-A, III-C, IV-C),c. (1-D, II-A, III-C, IV-B),d. (1-A, II-D, III-C, IV-B)Option (a) matches entirely with our correct pairing: 1 matched \(\text{Permittivity of free space}\) with D, III matched \(\text{Electrical potential}\) with B, IV matched \(\text{Electrical energy}\) with C.
Key Concepts
Electrical CapacityPermittivity of Free SpaceElectrical PotentialElectrical Energy
Electrical Capacity
Electrical capacity, also known as capacitance, is the ability of a system to store electric charge. Capacitors are components that exhibit electrical capacity.
The standard unit of measurement for electrical capacity is the farad (F), named after Michael Faraday. The dimensional formula for capacitance according to SI units is \([\text{M}^{-1} \text{L}^{-2} \text{T}^4 \text{A}^2]\). However, this is not provided in the original Column II options.
A capacitor stores energy in the form of an electric field, created by the separation of charges onto different plates. The larger the capacity, the more charge it can store for a given voltage difference between its plates. It's important to understand how capacitors work, as they are fundamental components in electronic circuits, enabling the smooth operation of devices by storing and releasing energy as needed.
The standard unit of measurement for electrical capacity is the farad (F), named after Michael Faraday. The dimensional formula for capacitance according to SI units is \([\text{M}^{-1} \text{L}^{-2} \text{T}^4 \text{A}^2]\). However, this is not provided in the original Column II options.
A capacitor stores energy in the form of an electric field, created by the separation of charges onto different plates. The larger the capacity, the more charge it can store for a given voltage difference between its plates. It's important to understand how capacitors work, as they are fundamental components in electronic circuits, enabling the smooth operation of devices by storing and releasing energy as needed.
Permittivity of Free Space
Permittivity of free space, represented by the symbol \(\varepsilon_0\), is a constant that characterizes the ability of the vacuum (or space) to allow electric field lines to pass through. It is a cornerstone in the equations of electromagnetism, especially in Coulomb's Law and Gauss's Law.
The dimensional formula for permittivity of free space is \([\text{M}^{-1} \text{L}^{-3} \text{T}^4 \text{A}^2]\), which matches the option D given in Column II. This value is crucial in defining the strength of interactions between electric charges in a vacuum.
In simple terms, permittivity determines how much resistance is encountered when forming an electric field in a medium. Higher permittivity means less resistance and a stronger capacity to store additional charge. This concept is essential for understanding how electric fields behave in different environments, whether in a vacuum or within various materials.
The dimensional formula for permittivity of free space is \([\text{M}^{-1} \text{L}^{-3} \text{T}^4 \text{A}^2]\), which matches the option D given in Column II. This value is crucial in defining the strength of interactions between electric charges in a vacuum.
In simple terms, permittivity determines how much resistance is encountered when forming an electric field in a medium. Higher permittivity means less resistance and a stronger capacity to store additional charge. This concept is essential for understanding how electric fields behave in different environments, whether in a vacuum or within various materials.
Electrical Potential
Electrical Potential, also known as voltage, measures the potential energy per unit charge. It indicates how much work would be needed to move a charge between two points within an electric field. This measure is fundamental in understanding how electric fields do work and exert force on charges.
The SI unit for electrical potential is the volt (V), and its dimensional formula is \([\text{M}^1 \text{L}^2 \text{T}^{-3} \text{A}^{-1}]\), which corresponds with option B in Column II.
Voltage is the driving force behind the movement of charges in circuits. It works like pressure that pushes electrons through a conductor, allowing electricity to perform work. Devices like batteries maintain a potential difference to ensure ongoing current flow, powering everything from small gadgets to entire buildings.
The SI unit for electrical potential is the volt (V), and its dimensional formula is \([\text{M}^1 \text{L}^2 \text{T}^{-3} \text{A}^{-1}]\), which corresponds with option B in Column II.
Voltage is the driving force behind the movement of charges in circuits. It works like pressure that pushes electrons through a conductor, allowing electricity to perform work. Devices like batteries maintain a potential difference to ensure ongoing current flow, powering everything from small gadgets to entire buildings.
Electrical Energy
Electrical energy refers to the energy carried by moving charges. It's a form of energy resulting from the interaction of charged particles, usually within a conductor.
The dimensional formula for electrical energy is \([\text{M}^1 \text{L}^2 \text{T}^{-2}]\), matching option C from Column II.
Electrical energy is calculated by multiplying electrical potential by the charge moved (in joules). It is the bedrock of all electrical work, powering lights, appliances, machinery, and electronic devices. Understanding how electrical energy is harnessed and transmitted allows us to optimize power systems and manage energy consumption more efficiently.
Examples where electrical energy is essential include household electrical systems, electric vehicles, and even large-scale power grids that distribute energy from power plants to homes and businesses.
The dimensional formula for electrical energy is \([\text{M}^1 \text{L}^2 \text{T}^{-2}]\), matching option C from Column II.
Electrical energy is calculated by multiplying electrical potential by the charge moved (in joules). It is the bedrock of all electrical work, powering lights, appliances, machinery, and electronic devices. Understanding how electrical energy is harnessed and transmitted allows us to optimize power systems and manage energy consumption more efficiently.
Examples where electrical energy is essential include household electrical systems, electric vehicles, and even large-scale power grids that distribute energy from power plants to homes and businesses.
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