Question

A physical quantity is represented by $P = X^aY^bZ^c$. If the errors in the measurements of the physical quantities X, Y and Z are respectively a%, b% and c%, then the percentage error in the determination of quantity P is

• A. $a^2+b^2+c^2$
• B. $a+b+c$
• C. $a^2+b^2-c^2$
• D. $a+b-c$

Hint:

The variable names for the percentage errors ($a, b, c$) happen to be the same as the exponents in this specific problem. Don't get confused! Usually, the formula is: $%P = (\text{power}_x)(%x) + (\text{power}_y)(%y) + \dots$ Here, power is 'a' and error is 'a', so we get $a \cdot a = a^2$.

Answer 1

The Correct Option is A

Step 1: Understanding Error Propagation:
For any physical quantity $P$ defined by the relation $P = X^a Y^b Z^c$, the maximum relative error is calculated by summing the products of the exponents and the relative errors of the individual quantities. The general formula is: \[ \frac{\Delta P}{P} = a \left( \frac{\Delta X}{X} \right) + b \left( \frac{\Delta Y}{Y} \right) + c \left( \frac{\Delta Z}{Z} \right) \] (Note: Even if an exponent is negative in the formula, we take absolute values because we calculate the \maximum possible error).
Step 2: Substituting Given Values:
The question gives the percentage errors for X, Y, and Z. Percentage error is defined as $\frac{\Delta Q}{Q} \times 100$. Given:

• $%$ Error in $X = \frac{\Delta X}{X} \times 100 = a$
• $%$ Error in $Y = \frac{\Delta Y}{Y} \times 100 = b$
• $%$ Error in $Z = \frac{\Delta Z}{Z} \times 100 = c$

Step 3: Calculation:
We want the percentage error in P, which is $\frac{\Delta P}{P} \times 100$. Multiply the error propagation equation by 100: \[ \frac{\Delta P}{P} \times 100 = a \left( \frac{\Delta X}{X} \times 100 \right) + b \left( \frac{\Delta Y}{Y} \times 100 \right) + c \left( \frac{\Delta Z}{Z} \times 100 \right) \] Substitute the given percentage values ($a, b, c$) into the equation: \[ % \text{ Error in } P = a(a) + b(b) + c(c) \] \[ % \text{ Error in } P = a^2 + b^2 + c^2 \]
Step 4: Final Answer:
The percentage error in P is $a^2 + b^2 + c^2$.