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## Homework Statement

Prove the following equation:

## \Delta U=\frac {R_1R_4}{(R_1+R_4)^2}(\frac {\Delta R_1}{R_1}-\frac {\Delta R_2}{R_2}+\frac{\Delta R_3}{R_3}-\frac{\Delta R_4}{R_4})E##

This is used in Wheatstone bridge

## Homework Equations

[/B]

U=RI

## The Attempt at a Solution

This has been a real head-scratcher

Two voltage dividers can be found for starters. Voltage's direction is assumed to be clockwise

##V_{in1}=I_2(R_2+R_3)##

##I_2=\frac{V_{in1}}{R_2+R_3}##

##V_{out1}=I_2R_3##

##V_{out1}=V_{in1}\frac{R_3}{R_2+R_3}##

Similarly:

##V_{out2}=V_{in1}\frac{R_4}{R_1+R_4}##

##V_G## is voltage between A and B

##V_{out1}-V_{out2}=V_G##

##V_{in1}\frac{R_3}{R_2+R_3}-V_{in1}\frac{R_4}{R_1+R_4}=V_G##

##V_{in1}(\frac{R_3}{R_2+R_3}-\frac{R_4}{R_1+R_4})=V_G##

##V_{in1}=E##

##V_G=\Delta U## so then

##E(\frac{R_3}{R_2+R_3}-\frac{R_4}{R_1+R_4})=\Delta U##

I have calculated voltages in different circuits and tried to think this problem in different ways, but the real problem is that how is ##\Delta R_i## inserted into equations. Assumption goes that it is added by ##R_i+\Delta R_i##. Maybe that is incorrect?

Help is very much appreciated!

edit: Misspelling corrected

Also particularizing that ##\Delta R_i## is a change in one resistance

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