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A power dissipated in a resister is given by p= E²/R using calculus, find R the approximate percentage error in P, when E is increased by 3% and R is decreased by 2%?
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A power dissipated in a resister is given by p= E²/R using calculus, f...
Understanding Power Dissipation in a Resistor
The power (P) dissipated in a resistor can be expressed as:
P = E² / R
Where:
- E = Voltage across the resistor
- R = Resistance
Calculating Percentage Changes
To find the approximate percentage error in power (P) when E is increased and R is decreased, we first consider the percentage changes:
- E is increased by 3%, so the relative change in E is +0.03.
- R is decreased by 2%, so the relative change in R is -0.02.
Using Differentiation for Approximation
We apply calculus (differentiation) to understand how changes in E and R affect P:
1. The differential of P with respect to E is:
dP/dE = 2E/R
2. The differential of P with respect to R is:
dP/dR = -E²/R²
Using the chain rule, the total differential dP can be expressed as:
dP = (dP/dE) dE + (dP/dR) dR
Substituting the relative changes:
- dE/E = 0.03
- dR/R = -0.02
Calculating Total Percentage Change in P
The total relative change in P is:
dP/P = (dP/dE)(dE/E) + (dP/dR)(dR/R)
Substituting the values, we get:
dP/P ≈ (2E/R)(0.03) + (-E²/R²)(-0.02)
This simplifies to:
dP/P ≈ 0.06 + 0.02 = 0.08
Final Percentage Error in Power (P)
Thus, the approximate percentage error in P is:
- 8% increase in power when E is increased by 3% and R is decreased by 2%.
This analysis highlights the significant impact of changes in voltage and resistance on power dissipation in a resistor.
The power (P) dissipated in a resistor can be expressed as:
P = E² / R
Where:
- E = Voltage across the resistor
- R = Resistance
Calculating Percentage Changes
To find the approximate percentage error in power (P) when E is increased and R is decreased, we first consider the percentage changes:
- E is increased by 3%, so the relative change in E is +0.03.
- R is decreased by 2%, so the relative change in R is -0.02.
Using Differentiation for Approximation
We apply calculus (differentiation) to understand how changes in E and R affect P:
1. The differential of P with respect to E is:
dP/dE = 2E/R
2. The differential of P with respect to R is:
dP/dR = -E²/R²
Using the chain rule, the total differential dP can be expressed as:
dP = (dP/dE) dE + (dP/dR) dR
Substituting the relative changes:
- dE/E = 0.03
- dR/R = -0.02
Calculating Total Percentage Change in P
The total relative change in P is:
dP/P = (dP/dE)(dE/E) + (dP/dR)(dR/R)
Substituting the values, we get:
dP/P ≈ (2E/R)(0.03) + (-E²/R²)(-0.02)
This simplifies to:
dP/P ≈ 0.06 + 0.02 = 0.08
Final Percentage Error in Power (P)
Thus, the approximate percentage error in P is:
- 8% increase in power when E is increased by 3% and R is decreased by 2%.
This analysis highlights the significant impact of changes in voltage and resistance on power dissipation in a resistor.
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A power dissipated in a resister is given by p= E²/R using calculus, find R the approximate percentage error in P, when E is increased by 3% and R is decreased by 2%?
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A power dissipated in a resister is given by p= E²/R using calculus, find R the approximate percentage error in P, when E is increased by 3% and R is decreased by 2%? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A power dissipated in a resister is given by p= E²/R using calculus, find R the approximate percentage error in P, when E is increased by 3% and R is decreased by 2%? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A power dissipated in a resister is given by p= E²/R using calculus, find R the approximate percentage error in P, when E is increased by 3% and R is decreased by 2%?.
A power dissipated in a resister is given by p= E²/R using calculus, find R the approximate percentage error in P, when E is increased by 3% and R is decreased by 2%? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A power dissipated in a resister is given by p= E²/R using calculus, find R the approximate percentage error in P, when E is increased by 3% and R is decreased by 2%? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A power dissipated in a resister is given by p= E²/R using calculus, find R the approximate percentage error in P, when E is increased by 3% and R is decreased by 2%?.
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