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Chapter 1: Problem 127

Which physical quantity is represented by \(\mathrm{M}^{1} \mathrm{~L}^{3} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\) ? (a) Resistivity (b) Resistance (c) conductance (d) conductivity

Short Answer

Expert verified
The given dimensional formula \(\mathrm{M^{1}L^{3}T^{-3}A^{-2}}\) does not represent any of the given physical quantities (resistivity, resistance, conductance, or conductivity).

Step by step solution

01

(a) Resistivity #

The dimensional formula for resistivity, denoted by \(\rho\), is: $$\rho = \frac{\mathrm{M}^{0} \mathrm{L}^{3} \mathrm{T}^{-3} \mathrm{A}^{-2}}{\mathrm{M}^{0} \mathrm{L}^{1} \mathrm{T}^{0} \mathrm{A}^{0}} = \mathrm{M}^{0} \mathrm{L}^{3-1} \mathrm{T}^{-3} \mathrm{A}^{-2} = \mathrm{M}^{0} \mathrm{L}^{2} \mathrm{T}^{-3} \mathrm{A}^{-2}$$
02

(b) Resistance #

The dimensional formula for resistance, denoted by R, is: $$R = \frac{\mathrm{M}^{0} \mathrm{L}^{3} \mathrm{T}^{-3} \mathrm{A}^{-2}}{\mathrm{M}^{0} \mathrm{L}^{2} \mathrm{T}^{0} \mathrm{A}^{0}} = \mathrm{M}^{0} \mathrm{L}^{3-2} \mathrm{T}^{-3} \mathrm{A}^{-2} = \mathrm{M}^{0} \mathrm{L}^{1} \mathrm{T}^{-3} \mathrm{A}^{-2}$$
03

(c) Conductance #

The dimensional formula for conductance, denoted by G, is: $$ G = \frac{1}{R} \implies G = \mathrm{M}^{0} \mathrm{L}^{-1} \mathrm{T}^{3} \mathrm{A}^{2}$$
04

(d) Conductivity #

The dimensional formula for conductivity, denoted by \(\sigma\), is: $$ \sigma = \frac{1}{\rho} \implies \sigma = \mathrm{M}^{0} \mathrm{L}^{-2} \mathrm{T}^{3} \mathrm{A}^{2} $$ # Step 2: Compare the given formula with each option # Now, we will compare the given dimensional formula \(\mathrm{M^{1}L^{3}T^{-3}A^{-2}}\) with the dimensional formulas of the given options. _resistivity_: \(\mathrm{M^{0}L^{2}T^{-3}A^{-2}}\) _resistance_: \(\mathrm{M^{0}L^{1}T^{-3}A^{-2}}\) _conductance_: \(\mathrm{M^{0}L^{-1}T^{3}A^{2}}\) _conductivity_: \(\mathrm{M^{0}L^{-2}T^{3}A^{2}}\) None of the given options match with the dimensional formula \(\mathrm{M^{1}L^{3}T^{-3}A^{-2}}\). Therefore, the given dimensional formula does not represent any of the given physical quantities (resistivity, resistance, conductance, or conductivity).

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Key Concepts

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

Resistivity
Resistivity is a fundamental property of materials that quantifies how strongly a material opposes the flow of electric current. It is denoted by the symbol \( \rho \) and has the dimensional formula \( \mathrm{M}^{0} \mathrm{L}^{3} \mathrm{T}^{-3} \mathrm{A}^{-2} \div \mathrm{M}^{0} \mathrm{L}^{1} \mathrm{T}^{0} \mathrm{A}^{0} = \mathrm{M}^{0} \mathrm{L}^{2} \mathrm{T}^{-3} \mathrm{A}^{-2} \). Breaking this down:
  • Length (L): Indicates the material's structure affects its resistance to current.
  • Time (T): Shows the dependence on time-related factors like electron movement.
  • Current (A): Represents the amount of electrical charge.
In practical terms, high resistivity means a material is a poor conductor since it resists the flow of electric current well.
Resistance
Resistance is the measure of how much a conductor opposes the flow of electric current. Symbolized by \( R \), its dimensional formula is \( \mathrm{M}^{0} \mathrm{L}^{1} \mathrm{T}^{-3} \mathrm{A}^{-2} \).
  • Length (L): The bulk value contributes to the physical pathway's impact on current flow.
  • Current (A): Relates to how electrical charge interacts with the conductor.
When you place a material across a voltage difference, resistance determines how much the current is reduced, following Ohm's Law: \( V = IR \). Greater resistance implies less current for the same voltage difference.
Conductance
Conductance is essentially the reciprocal of resistance, denoted by \( G \). Its dimensional formula is \( \mathrm{M}^{0} \mathrm{L}^{-1} \mathrm{T}^{3} \mathrm{A}^{2} \). This property measures the ease with which electric current can flow through a material. Here are a few key points:
  • Length (L): Negative dimension indicates it describes a reduction in area-specific opposition.
  • Current (A): A higher conductance means more electric charge can pass.
  • Time (T): Suggests a relation to time-domain properties like response time.
Conductance is integral in designing circuits, as high conductance implies better performance for efficient current flow.
Conductivity
Conductivity, denoted by \( \sigma \), measures a material's ability to facilitate the flow of electric current, being the inverse of resistivity. Its dimensional formula is \( \mathrm{M}^{0} \mathrm{L}^{-2} \mathrm{T}^{3} \mathrm{A}^{2} \). Conductivity features:
  • Length (L): Reflects direct contribution to the flow rate of current, as seen in area-specific measures.
  • Time (T): Communicates response time and dependencies on time-varying conditions.
  • Current (A): Superior conductivity means carriers like electrons move efficiently.
In metals, high conductivity is common, explaining why metals are preferred for wiring and components needing excellent electrical performance.

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