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Question

A student uses dimensional analysis to analyze the time ($	au$) it takes for a capacitor (C) to discharge through a resistor (R). They correctly identify that $	au$ depends on R and C.  However, they are unable to derive the precise relationship $	au = RC$. What limitation of dimensional analysis explains this?

P.NARASIMHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: Option A
## CONCEPT FRAMEWORK
**Primary Concept**: Dimensional Analysis
**Related Topics**: Units, Physical Quantities, Constants
**Prerequisites**: Understanding of dimensional analysis principles
## SOLUTION PATHWAY
**Given Information**:
* Time constant ($	au$) depends on resistance (R) and capacitance (C).
* Goal is to understand why dimensional analysis fails to give the exact relation $	au = RC$
**Method & Steps**:
1. Dimensional analysis can only determine the relationship between the dimensions of the variables. It cannot determine the numerical constants.
   Reasoning: Dimensional analysis focuses on the fundamental units (like mass, length, time, charge, etc.) and how they relate in an equation.  It doesn't provide information about dimensionless numerical constants or coefficients.
   Formula/Rule: Dimensional analysis uses the fundamental dimensions of the physical quantities
2. The time constant $	au$ has dimensions of time [T]. Resistance R has dimensions of [V/I] or [ML^2T^-3I^-2]. Capacitance C has dimensions of [Q/V] or [M^-1L^-2T^4I^2].  Multiplying R and C gives [ML^2T^-3I^-2][M^-1L^-2T^4I^2] = [T].
   Working: The dimensional analysis will show that $	au$ is proportional to RC, i.e., $	au \propto R^a C^b$. By comparing dimensions, we get a=1, b=1. Thus, we have  $	au \propto RC$. However, it cannot tell us that the proportionality constant is 1.
   Key Point: Dimensional analysis can only determine the powers of the variables, but it cannot determine the numerical coefficient.
3. The correct relationship is $	au = RC$, but dimensional analysis can only show that $	au$ has the dimensions of RC, which means that $	au \propto RC$.  The numerical constant '1' is not found via dimensional analysis.
   Result: Dimensional analysis cannot reveal the precise relationship because it can't find numerical constants.
   Verification: The relationship between $	au$, R, and C is not just dimensional, but also includes a numerical factor.
## OPTION ANALYSIS
[A] ✓: Dimensional analysis can only determine the powers of the variables, but it cannot determine the numerical coefficient. This is why it cannot reveal the exact relationship $	au=RC$. The constant 1 is not available from dimensions.
[B] :x: The initial charge on the capacitor does affect the amount of charge discharged, but not the time constant. Dimensional analysis is concerned with the dimensions of the variables and not their initial values.
[C] :x: While exponential decay is involved, the relationship between the time constant and R and C is not an exponential function itself. Dimensional analysis is about the dimensions, not the form of the function.
[D] :x: The relationship between $	au$, R, and C is linear, not logarithmic. Dimensional analysis does not determine the form of a function, but only the dimensional consistency of the equation.
## LEARNING AIDS
**Common Mistakes**:
1. [Error 1]: Assuming dimensional analysis gives the exact relationship. It only gives the relationship up to a numerical constant.
   [How to avoid]: Remember that dimensional analysis only tells how the dimensions of variables relate and not the numerical constants.
2. [Error 2]: Confusing dimensional analysis with an exact solution. Dimensional analysis is a preliminary tool.
   [How to avoid]: Always treat dimensional analysis as a tool for analyzing the consistency of dimensions, not as a final tool for determining the exact relationship.
**Quick Tips**:
1. [Helpful shortcut]: Dimensional analysis is a good way to verify if an equation makes sense. If the dimensions don't match, the equation is wrong.
2. [Memory aid]: Remember that dimensional analysis can only determine the powers of the variables but not the numerical constants.

P.NARASIMHA · Answer

The correct Option is "Dimensional analysis cannot reveal purely numerical relationships between variables."

P.NARASIMHA · Answer

The analysis requires knowledge of the initial charge on the capacitor, which is not provided.

P.NARASIMHA · Answer

Dimensional analysis cannot handle exponential decay processes.

P.NARASIMHA · Answer

The relationship between $	au$, R, and C is logarithmic, not a power law.

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