Question

A potentiometer wire has a resistance 40 $\Omega$ and its length is 10 m. it is connected to a resistance of 760 $\Omega$ in series. If emf of battery is 2 V, then potential gradient is:
A) $0.5 \times 10^-6 V / m$
B) $1 \times 10^-6 V / m$
C) $1 \times 10^-2 V / m$
D) $2 \times 10^-6 V / m$

Answer 1

We will first find out the total resistance of the setup using the formula of series combination of the resistances: $R = {R_1} + {R_2}$ and then we will find out the resistance per unit length of the potentiometer wire by dividing the resistance of the wire with the length of the wire. Next, we will find out the current I using the relation: I = $\dfrac{V}{R}$. In the end, we need to find out the potential gradient by the formula: current $\times$ resistance per unit length.

Compete step by step solution
We are given a potentiometer wire. Its resistance is 40$\Omega$. It is connected to a resistance of 760$\Omega$ in series.
So, the total resistance of the setup, by the formula of series combination, can be calculated as:
$\Rightarrow R = {R_1} + {R_2}$
$\Rightarrow R = 40 + 760 = 800\Omega$
We know the resistance of the wire and the length of the wire as well. So, we can calculate the resistance per unit length of the wire.
$\Rightarrow$ resistance per unit length of the wire = resistance of the wirelength of the wire
$\Rightarrow$resistance per unit of the wire = $\dfrac{{40}}{{10}} = 4\Omega$
The emf of the battery is given as 2 V. so, we can calculate the current flowing through the circuit using: I = $\dfrac{V}{R}$
$\Rightarrow I = \dfrac{2}{{800}} = \dfrac{1}{{400}}A$
We are required to do the potential gradient. We can calculate it by the formula:
Potential gradient = current $\times$ resistance per unit length
$\Rightarrow$potential gradient =$\dfrac{1}{{400}} \times 4 = \dfrac{1}{{100}} = 1 \times {10^{ - 2}}V/m$.
Therefore, option(C) is correct.

Note:
You may get confused because of the variety of the formulae used in this question. You can not calculate the current flowing through the whole circuit just by using the value of resistance of the wire. So, you need to find the resistance of the whole combination and then only you can calculate the current because the current is flowing in the complete circuit.