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The length of the potentiometer wire is 10 m and is associated in series with an accumulator. The emf of a cell balances against $250\;{\text{cm}}$ length of the wire. If the length of the potentiometer wire is increased by $1\;{\text{m}}$, calculate the new balancing length of wire.

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Last updated date: 23rd Feb 2024
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Hint: The above problem can be solved by using the principle of the potentiometer. The potentiometer is the instrument that measures the internal resistance and emf of the unknown electrical source. The resistance of the electrical source varies with the length of the wire connected to the potentiometer.

Complete step by step answer:
Given: The length of the potentiometer wire is $l = 10\;{\text{m}}$, the length of the wire that balances the emf of the cell is ${l_b} = 250\;{\text{cm}} = 250\;{\text{cm}} \times \dfrac{{1\;{\text{m}}}}{{100\;{\text{cm}}}} = 2.50\;{\text{m}}$, the increase in the length of the potentiometer wire is $\Delta l = 1\;{\text{m}}$.
The expression to calculate the new balancing length of the wire is given as:
$L = \left( {\dfrac{{l + \Delta l}}{l}} \right){l_b}......\left( 1 \right)$
Substitute .$10\;{\text{m}}$. for $l$ , 1 m for $\Delta l$and $2.50\;{\text{m}}$ for ${l_b}$ in the expression (1) to calculate the length of new balancing wire.
$L = \left( {\dfrac{{10\;{\text{m}} + 1\;{\text{m}}}}{{10\;{\text{m}}}}} \right)\left( {2.5\;{\text{m}}} \right)$
$L = 2.75\;{\text{m}}$

Thus, the length of the new balancing wire is $2.75\;{\text{m}}$.

Additional Information: The wire of known resistance is always connected in the potentiometer. A resistance of variable resistance and variable length is connected to the galvanometer. The length of the variable resistance wire changes until the galvanometer does not show no deflection reading. The length of the variable resistance wire at which the galvanometer reading becomes zero is called the balancing length of the wire.

Note: Be careful in substituting the values of the length of known wire, unknown wire and balancing length of the wire. First convert all the units in the same measurement system. The internal resistance of the unknown source can be found by using following formula:
$r = R\left( {\dfrac{\varepsilon }{V} - 1} \right)$
Here, r is the internal resistance of the unknown source, R is the internal resistance of the known source, $\varepsilon $ is the emf of the known source, V is the potential difference across the known length of the wire.