Question

The wire in a potentiometer has a resistance of r0 and the potentiometer is connected to a battery of voltage v .now a resistor r whose value of resistance has to be measured is connected . when the sliding point is exactly in the middle of the potentiometer , the voltage drop across r is v/4 .what is the value of r/r0 ?

Answer 1

Answer:

Ratio of the given resistors is

$\frac{r}{r_o} = \frac{1}{4}$

Explanation:

As we know that the sliding point when reached at mid point of potentiometer  then the voltage drop across the given resistor is V/4

So here we can say that the resistance r and half of ro resistance in parallel so total resistance here is given as

$r_{eq} = \frac{\frac{r_o}{2} r}{\frac{r_o}{2} + r}$

Since the voltage drop across the resistance r is given V/4 so the voltage drop on the remaining half part is 3V/4

So by ohm's law we can say

V = i R

Now for two parts we have

$\frac{3V}{4} = i \frac{r_o}{2}$

and we also have

$\frac{V}{4} = i\frac{r r_o}{r_o + 2r}$

so from above two equations we have

$3 = \frac{r_o + 2r}{2r}$

$6r = r_o + 2r$

so we have

$\frac{r}{r_o} = \frac{1}{4}$

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Topic : Potentiometer

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Answer 2

Given :   The wire in the potentiometer has a resistance of R₀ and the potentiometer is connected to a battery of voltage 'V'.

Now a resistor 'R' whose value of resistance has to be measured is connected.  

When the sliding point is exactly in the middle of the potentiometer, the voltage drop across 'R' is V/4.

To Find :  the value of R/R₀

Solution:

sliding point is exactly in the middle of the potentiometer

Hence  potentiometer  resistance  =  R₀ /2

a resistor 'R'  

Hence total resistance  1/ Rₓ  = 1/R  +  1/(R₀ /2)

=> 1/Rₓ  = 1/R  + 2/R₀

=> 1/Rₓ  =   (R₀ + 2R)/RR₀

=> Rₓ = RR₀/(R₀ + 2R)

Total Rt  =  Rₓ   +  R₀ /2

=> Rt =  RR₀/(R₀ + 2R)  + R₀ /2

=> Rt =  (2RR₀  +R₀² + 2R R₀) /2(R₀ + 2R)  

=> Rt =  ( R₀² + 4R R₀) /2(R₀ + 2R)  

Hence current  =   V/ Rt

=> I   =   2V(R₀ + 2R) /  ( R₀² + 4R R₀)  

V/4  =   I  Rₓ

=> V/4  = Rₓ 2V(R₀ + 2R) /  ( R₀² + 4R R₀)  

=> ( R₀² + 4R R₀)    = 8 Rₓ (R₀ + 2R)

Rₓ = RR₀/(R₀ + 2R)

=>  ( R₀² + 4R R₀)    = 8  RR₀

=> R₀ + 4R  =  8R

=> R₀  = 4R

=> R/R₀  = 1/4

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