Potentiometer

The potentiometer or slide wire potentiometer is a device that is used to measure the emf of a cell without drawing any current from it. It can also be used to determine the internal resistance of the cell and to compare the unknown emf two or more cells. It was invented by Poggendorf in 1850.

Construction: It consists of a wooden board on which ten (sometimes five) wires of uniform cross-sectional area, low temperature coefficient of resistance and high resistivity like manganin, constantan etc. are stretched parallel to each other. Each wire is $1$ meter long. They are joined by thick copper strips so the combination behaves like a single wire of length $10$ meters.

Potentiometer

The end $A$ of the potentiometer wire is connected to the positive terminal of driving cell $E_d$ and negative terminal to the other end $B$. The jockey is slide along the wire $AB$ to get the balancing point $C$. The potentiometer is said to be balanced if there is no deflection in the galvanometer.

Principle: It is based on the principle that when constant current is passed through a wire of uniform cross sectional area, the potential difference across any segment of the wire is directly proportional to its length.

\[V ∝ l \] \[\therefore\frac{V_1}{V_2}=\frac{l_1}{l_2}\]

Potentiometer is an ideal voltmeter.

An ideal voltmeter is a voltmeter which does not change the original potential difference between any two points across which it is connected for which the resistance of the voltmeter should be infinite. Potentiometer is said to be an ideal voltmeter because It is based on null deflection method so it does not draw any current from the given circuit and still measures the potential difference.

To obtain the null deflection, the emf of driving cell must be greater than the emf of the experimental cell. A long wire is used to obtain small potential gradient so that accuracy increases.

Measurement of Internal Resistance of a Cell

For the determination of internal resistance of a cell, the following connections are made.

Using Potentiometer to Determine the Internal Resistance of a Cell

Let $E$ be the emf, $r$ be the internal resistance and $V$ be the terminal p.d. across the terminals of the experimental cell when supplying a current I through the external resistance $R$ (Resistance Box R.B.) by a driving cell of $E_d$. Then, from circuit formula, \[r=\left(\frac{E-V}{V}\right)R\;\;\text{__(1)}\]

Firstly, the key $k$ is kept open and the jockey $J$ is slided along the wire to get balancing point $C$. Let $l_1$ be the balancing length corresponding to the balancing point $C$. In this condition, emf of the cell is balanced by potential difference across $AC$ i.e. \[E=V_{AC}\]

According to the principle of potentiometer, \[V_{AC} ∝ l_1\] \[\text{i.e.} \;\; E ∝ l_1\] \[E=kl1 \text{ __(2)}\]

Secondly, the key $k$ is closed and a resistance $R$ is taken out from the resistance box (R.B.). The jockey $J$ is slided along the potentiometer wire to get another balancing point $D$. Let $l_2$ be the balancing length corresponding to the balancing point $D$. In this condition, the cell will give out current in the circuit and hence, terminal potential difference $(V)$ of the cell is balanced by potential difference across $AD$ i.e. \[V=V_{AD}\]

According to the principle of potentiometer, \[V_{AD} ∝ l_2\]

\[\text{i.e.}\;\; V ∝ l_2\] \[V=kl_2\;\;\text{__(3)}\]

From $(1)$, $(2)$ and $(3)$, \[r=\left(\frac{kl_1-kl_2}{kl_2}\right)R\] \[r=\left(\frac{l_1-l_2}{l_2}\right)R\]

By knowing the value of $R$, $l_1$ and $l_2$, the internal resistance $r$ of the cell can be calculated.


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