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Question

A resistance of R draws the current form of the potentiometer. The potentiometer wire, AB, has a total resistance of R0. A voltage V is supplied to the perpendicular. Derive an expression for the voltage across R when the sliding contact is in the middle of potentiometer wire.
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Solution

Given: A resistance of R draws the current form of the potentiometer. The potentiometer wire, AB, has a total resistance of R0. A voltage V is supplied to the perpendicular
To derive an expression for the voltage across R when the sliding contact is in the middle of potentiometer wire.
While the slide is in the middle of the potentiometer only half of its resistance, R02 will be between the points A and B. Hence, the total resistance between A and B, say. R1 will be given by the following expression:
1R1=1R+1R02R1=RR0R0+2R
The total resistance between A and C will be sum of the resistance between A and B and B and C, i.e., R1+R02
Therefore, the current flowing through the potentiometer will be
I=VR1+R02=2V2R1+R0
The voltage V1 taken from the potentiometer will be the product of current of I and resistance R1
V1=IR1=(2V2R1+R0)×R1
Substituting for R1, we have
V1=⎜ ⎜ ⎜2V2RR0R0+2R+R0⎟ ⎟ ⎟×RR0R0+2RV1=2VR2R+R0+2RV1=2VRR0+4R

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