Hello dear,

Hope you are doing well.

As per your question there are two potentiometer A and B of same cross section but unequal length.

Length of A = 10 m

Length of B = 5 m

You must know that potentiometers are used to measure unknown emf among the circuit by comparison with the known emf  and the sensitivity of the potentiometer depends upon the potential gradient.

Now, Potential gradient by its definition says the ratio of Voltage or potential to the length of the wire.

i.e, Potential Gradient = V/L

The potentiometer is said to be more sensitive if it measures the smallest voltage difference with more accurately.

From the above point we can say that the decrease in potential gradient will result in the increase in sensitivity of the potentiometer.

There are two ways we can decrease the potential gradient and are

1. Increasing the length of wire
2. Decreasing the current in the circuit if the length is fixed.

Point 1 says that by increasing the length of the wire the potential gradient i.e, voltage per unit length decreases and hence the device will be more sensible.

Since length of wire of A > length of wire of B, Potentiometer would be more sensible and we prefer potentiometer A for more accurate measurement.

I hope your query has been resolved and please feel free to connect in the comment section for further queries.

Happy helping.

Thank you!!!!