Calculating Error in Voltage (V) from Errors in I & R

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Discussion Overview

The discussion revolves around calculating the error in voltage (V) derived from known errors in current (I) and resistance (R) using the equation V=IR. Participants explore methods for error propagation and the applicability of certain formulas in different contexts.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant inquires about calculating the error in voltage given known errors in current and resistance, questioning whether a general method exists for any equation.
  • Another participant proposes a formula for calculating the relative error in voltage based on the relative errors in current and resistance, suggesting a square root approach.
  • Some participants discuss the bounds of voltage based on the maximum and minimum values derived from the uncertainties in I and R, questioning how to derive the error in voltage from these bounds.
  • There is a mention of a standard method for error propagation that can be applied to multiplication or division, with a participant affirming its validity based on their experience with error analysis.
  • One participant notes that the formula for error propagation requires the measurements to be independent and have normally distributed random errors, raising a caution about its applicability.
  • Another participant clarifies that the original problem implies known maximum possible errors rather than random errors, which adds complexity to the discussion.

Areas of Agreement / Disagreement

Participants generally agree on the standard method for error propagation in multiplication and division, but there is disagreement regarding the conditions under which the proposed formulas are valid, particularly concerning the nature of the errors involved.

Contextual Notes

Some limitations are noted regarding the assumptions of independence and normal distribution of errors, which may affect the applicability of the discussed formulas. Additionally, the discussion does not resolve the nuances of using maximum possible errors versus random errors.

Who May Find This Useful

This discussion may be useful for students and professionals engaged in experimental physics or engineering who are interested in error analysis and propagation techniques.

mathlete
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I don't know if this more of a math question than a physics question, but here goes:

Let's say I have an equation, let's say V=IR. Let's also say I know [tex]\sigma I[/tex] and [tex]\sigma R[/tex], the possible errors in I and R, how can I calculate [tex]\sigma V[/tex], the error in voltage? Is there a way to generalize this for any equation?
 
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square root of the sum of the error over the actual values squared equals error of V over V.

√((∆I/I)2 + (∆R/R)2 ) = ∆V/V

so ∆V = (V)√((∆I/I)2 + (∆R/R)2 )
 
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What you are saying, I think, is that the actual value of R lies between R-δR and R+ δR and that the actual value of I lies between I- δI and I+ δI.

The smallest possible value then of V is (R-δR)(I- δI)= RI- 2δRδI+ (δR)(δI) while the largerst possible value is (R+ δR)(I+ δI)= RI+ 2δRδI+ (δR)(δI).

Can you get the error in V from that?
 
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HallsofIvy said:
What you are saying, I think, is that the actual value of R lies between R-δR and R+ delta;R and that the actual value of I lies between I- δI and I+ δI.

The smallest possible value then of V is (R-δR)(I- δI)= RI- 2δRδI+ (δR)(δI) while the largerst possible value is (R+ δR)(I+ δI)= RI+ 2δRδI+ (δR)(δI).

Can you get the error in V from that?
Well I cheated a bit, I already KNEW the answer in the form Cantari gave it. I was wondering if that is the standard way to find error if you have an equation, by taking the square root of the sum of the errors of your components (I, R in this case) over the actual values squared?
 
Yes that is the standard way. Can be used for multiplication or division. Would be the same form if you wanted to find the error in I if you only had the errors in V and R. I = V/R.
 
Cantari said:
Yes that is the standard way. Can be used for multiplication or division. Would be the same form if you wanted to find the error in I if you only had the errors in V and R. I = V/R.

I did a lot of work with error propagation for a high school experiment earlier this year. Error analysis can be an extremely tedious task, especially when the problem involves many inter-related variables. The paper covers the error propagation in a complicated multistage chemical analysis. If you like I can email the pdf file to you or upload it to the net somewhere.
 
The uncertainty equation I gave is derived from a taylor expansion, it is indeed a proper way.
 
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The formula given by Cantari is not correct in every case. It requires that the measurements of the 2 quantities to be multiplied (a) are independent, and (b) have random errors with normal distributions.
 
But the original problem said that you KNOW the errors (which I took to mean maximum possible error), not that the errors were random.
 
  • #10
OK, the equations i'll need it for aren't very complex. They're all independent and the errors have normal distribution. Thanks for your help everyone.
 

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