Question

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

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

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:

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.

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.

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.

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.

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

Key concepts

Capacitance

Capacitance is a fundamental property of electrical components that defines their ability to store electric charge. It is particularly relevant in capacitors, which are devices designed to do just that. The capacitance of a component is determined by the equation:\[C = \frac{Q}{V}\]where \(C\) is the capacitance, \(Q\) is the charge stored in the device, and \(V\) is the voltage across it.
Capacitance is measured in farads (F), and its dimensional formula is given by \([M^{-1}L^{-2}T^4A^2]\). This formula arises because capacitance involves storing charge (related to current and time), and it inversely relates to potential difference or voltage.
Understanding capacitance is crucial in applications such as tuning circuits, filtering signals, and energy storage in electric power systems. It's also an essential concept in various electronic devices ranging from simple radio sets to complex computer memory storage technologies.

Resistance

Resistance is a measure of how much a material opposes the flow of electric current. It's a critical aspect of electrical circuits and plays a vital role in determining the efficiency and safety of electronic devices. The resistance of an object is calculated using Ohm's Law:\[R = \frac{V}{I}\]where \(R\) represents resistance, \(V\) is voltage, and \(I\) is current.
Resistance is measured in ohms (\(\Omega\)) and comes with the dimensional formula \([M^1L^2T^{-3}A^{-2}]\). This formula emerges because resistance involves electric current, voltage, and the material's physical properties, such as length and cross-sectional area of the conductor.
A resistor can control the current flow to safe levels, modulate signals in electronic devices, and convert electrical energy into heat in heaters and lamps. Recognizing the role of resistance is essential in designing circuits that can function efficiently without overheating or causing potential hazards.

Time Period

The time period of a wave refers to the time it takes to complete one full cycle of oscillation. This is a fundamental concept in oscillatory motion and wave mechanics, especially relevant in both mechanical and electronic oscillators. The time period is inversely related to frequency, the number of cycles per unit time:\[T = \frac{1}{f}\]where \(T\) represents the time period and \(f\) is the frequency.
The dimensional formula for the time period is \([T^1]\), indicating it purely represents time without involving mass, length, or electric charge. This simplicity is why it matches with the product of capacitance and resistance's dimensional formulas, as derived in the solved exercise.
Understanding the time period is crucial in various applications, such as calibrating clocks, designing communication signal waveforms, and optimizing performance in resonance phenomena across different engineering domains.