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Chapter 1: Problem 138

Which physical quantity has dimensional formula as \(\mathrm{CR}\) where \(\mathrm{C}\) - capacitance and \(\mathrm{R}\) - Resistance? (a) Frequency (b) current (c) Time period (d) acceleration

Short Answer

Expert verified
The physical quantity that has a dimensional formula represented as CR is the time period (T). Therefore, the correct answer is: (c) Time period

Step by step solution

01

Review the dimensions of capacitance and resistance

Before we check each option, let's recall the dimensions of capacitance (C) and resistance (R). \(Dimensional\ formula\ of\ capacitance (C)\): [M\(^{-1}\)L\(^{-2}\)T\(^4\)A\(^2\)] \(Dimensional\ formula\ of\ resistance (R)\): [M\(^1\)L\(^2\)T\(^{-3}\)A\(^{-2}\)] Now let's analyze each of the given options:
02

Check option (a) - Frequency

Frequency (f) is the number of oscillations (cycles) per unit time. The dimensional formula of frequency can be written as [T\(^{-1}\)]. Now, let's check if this matches CR: \(CR = [M\(^{-1}\)L\(^{-2}\)T\(^4\)A\(^2\)] × [M\(^1\)L\(^2\)T\(^{-3}\)A\(^{-2}\)] ≠ [T\(^{-1}\)] So, the dimensional formula for frequency is not CR.
03

Check option (b) - Current

The dimensional formula for current (I) is: [A\(^1\)]. Let's check if this matches CR: \(CR = [M\(^{-1}\)L\(^{-2}\)T\(^4\)A\(^2\)] × [M\(^1\)L\(^2\)T\(^{-3}\)A\(^{-2}\)] ≠ [A\(^1\)] So, the dimensional formula for current is not CR.
04

Check option (c) - Time period

The time period (T) is the time required for a wave to complete one cycle. Its dimensional formula is [T\(^1\)]. Let's check if this matches CR: \(CR = [M\(^{-1}\)L\(^{-2}\)T\(^4\)A\(^2\)] × [M\(^1\)L\(^2\)T\(^{-3}\)A\(^{-2}\)] = [T\(^1\)] Here, we can see that the dimensional formula of the time period (T) matches the product CR.
05

Check option (d) - Acceleration

It is not necessary to check option (d) since we have already found a match in option (c), but let's confirm for completeness: The dimensional formula for acceleration (a) is [LT\(^{-2}\)]. Let's check if this matches CR: \(CR = [M\(^{-1}\)L\(^{-2}\)T\(^4\)A\(^2\)] × [M\(^1\)L\(^2\)T\(^{-3}\)A\(^{-2}\)] ≠ [LT\(^{-2}\)] As expected, the dimensional formula for acceleration is not CR.
06

Conclude the answer

Based on our analysis, the physical quantity that has a dimensional formula represented as CR is the time period (T). Therefore, the correct answer is: (c) Time period

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