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A power dissipated in a resister is F2 given by p= E. using calculus, find R the approximate / error in P, when E is in creased by 3% and R is decreased by 2%?
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A power dissipated in a resister is F2 given by p= E. using calculus, ...
Understanding Power Dissipation in Resistors
The power dissipated in a resistor is given by the equation P = E/R, where P is power, E is voltage, and R is resistance. To analyze the effects of changes in E and R, we can apply calculus for small perturbations.
Calculating the Changes
1. Percentage Changes:
- E increases by 3%: ΔE = 0.03E
- R decreases by 2%: ΔR = -0.02R
2. Differential Approximation:
Using the formula for power, we can find the differential change in power:
dP = (∂P/∂E)dE + (∂P/∂R)dR
Here:
- ∂P/∂E = 1/R
- ∂P/∂R = -E/R^2
3. Substituting Changes:
- dP = (1/R)(0.03E) + (-E/R^2)(-0.02R)
- Simplifying, we get:
dP = (0.03E/R) + (0.02E/R) = (0.05E/R)
Calculating the Approximate Error in Power
The total change in power (dP) due to the small changes in E and R can be expressed as:
- dP ≈ 0.05 * (E/R)
Thus, the approximate increase in power is 5% of the original power P.
Conclusion
In summary, when E is increased by 3% and R is decreased by 2%, the power P increases by an approximate value of 5%. This analysis highlights the sensitivity of power dissipation in resistors with respect to changes in voltage and resistance.
The power dissipated in a resistor is given by the equation P = E/R, where P is power, E is voltage, and R is resistance. To analyze the effects of changes in E and R, we can apply calculus for small perturbations.
Calculating the Changes
1. Percentage Changes:
- E increases by 3%: ΔE = 0.03E
- R decreases by 2%: ΔR = -0.02R
2. Differential Approximation:
Using the formula for power, we can find the differential change in power:
dP = (∂P/∂E)dE + (∂P/∂R)dR
Here:
- ∂P/∂E = 1/R
- ∂P/∂R = -E/R^2
3. Substituting Changes:
- dP = (1/R)(0.03E) + (-E/R^2)(-0.02R)
- Simplifying, we get:
dP = (0.03E/R) + (0.02E/R) = (0.05E/R)
Calculating the Approximate Error in Power
The total change in power (dP) due to the small changes in E and R can be expressed as:
- dP ≈ 0.05 * (E/R)
Thus, the approximate increase in power is 5% of the original power P.
Conclusion
In summary, when E is increased by 3% and R is decreased by 2%, the power P increases by an approximate value of 5%. This analysis highlights the sensitivity of power dissipation in resistors with respect to changes in voltage and resistance.
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A power dissipated in a resister is F2 given by p= E. using calculus, find R the approximate / error in P, when E is in creased by 3% and R is decreased by 2%? for UPSC 2025 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 F2 given by p= E. using calculus, find R the approximate / error in P, when E is in creased by 3% and R is decreased by 2%? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A power dissipated in a resister is F2 given by p= E. using calculus, find R the approximate / error in P, when E is in creased by 3% and R is decreased by 2%?.
A power dissipated in a resister is F2 given by p= E. using calculus, find R the approximate / error in P, when E is in creased by 3% and R is decreased by 2%? for UPSC 2025 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 F2 given by p= E. using calculus, find R the approximate / error in P, when E is in creased by 3% and R is decreased by 2%? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A power dissipated in a resister is F2 given by p= E. using calculus, find R the approximate / error in P, when E is in creased by 3% and R is decreased by 2%?.
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