Question

A voltmeter reading up to $100mV$ is calibrated using a potentiometer. An EMF of $1.08V$ is balanced across $5.4m$ length of the wire. When voltmeter reading is $0.9V$, the balancing length is $4.55m$, the error in the voltmeter reading is

A. $0.01V$

B. $0.1V$

C. $0.001V$

D. $0V$

Answer 1

Hint: Potentiometer concept is to be used here which is that when a steady current flow through a wire of uniform cross section of potentiometer, the potential difference per unit length is constant throughout the length of wire

${{V}_{WIRE}} \propto \,{{l}_{wire}}$

$\dfrac{V}{L}$ of a potentiometer is constant and is known as voltage gradient.

Potentiometer here is used to find the error in voltage measured by voltmeter using this principle.

Complete step by step answer:

The emf given here is of the driving cell which drives the whole potentiometer and from which the voltage gradient of wire is to be found out.

Voltage gradient to be found out which when multiplied with balancing length gives actual voltage measured by the voltmeter.

Let $k$ be the potential gradient of the wire

$emf=1.08V$

$balancing\,length:5.4m$

$\Rightarrow k=\dfrac{1.08}{5.4}$

$\Rightarrow k=0.2........\left( 1 \right)$

It is a constant value

When voltmeter reading is found out to be $V=0.9Volts$,the balancing length was found to be ${{l}^{'}}=4.55m$.

$\Rightarrow {{V}_{actual}}=k\times {{l}^{'}}$

From $\left( 1 \right)$

$\Rightarrow {{V}_{actual}}=0.2\times 4.55\,Volts$

$\Rightarrow {{V}_{actual}}=0.91Volts$

Now the error in the reading is to be calculated which is 

$error={{V}_{actual}}-V$

$error=0.91-0.9Volts$

$error=0.01Volts$

So we have the error in the reading of voltmeter and the correct answer is  option (A).

Note:

The sensitivity of the potentiometer can be increased by decreasing the voltage gradient which can be done by increasing the number of wires in the potentiometer.

It can even be used to compare emf ’s of two cells.

Also the main point to note is that the emf of driving cell of potentiometer has to be greater than the emf of the cell to be measured for working of potentiometer.