Question

A battery of \(2 \mathrm{~V}\) and internal resistance 1 is used to send a current through a potentiometer wire of length \(200 \mathrm{~cm}\) and resistance \(4 \mathrm{Q}\) What is the potential gradient of the wire? (A) \(8 \times 10^{-3} \mathrm{~V} \mathrm{~cm}^{-1}\) (B) \(4 \times 10^{-3} \mathrm{~V} \mathrm{~cm}^{-1}\) (C) \(6 \times 10^{-3} \mathrm{~V} \mathrm{~cm}^{-1}\) (D) \(2 \times 10^{-3} \mathrm{~V} \mathrm{~cm}^{-1}\)

Short answer

The potential gradient of the wire is \(8 \times 10^{-3}~V~cm^{-1}\).

Step-by-step solution

Identify given values from the problem

The exercise provides us with the following information: - The battery has a voltage of \(2~V\) and an internal resistance of \(1~\Omega\). - The potentiometer wire has a length of \(200~cm\) (or \(2~m\)) and a resistance of \(4~\Omega\).

Calculate the total resistance of the system

The total resistance of the system is the sum of the internal resistance of the battery and the resistance of the potentiometer wire: \[R_{total} = R_{battery} + R_{wire}\] \[R_{total} = 1~\Omega + 4~\Omega\] \[R_{total} = 5~\Omega\]

Determine the current flowing through the system

Using Ohm's law, calculate the current flowing through the system (I): \[V = I \cdot R\] \[I = \frac{V}{R_{total}}\] Substitute the values of V and \(R_{total}\): \[I = \frac{2~V}{5~\Omega}\] \[I = 0.4~A\]

Calculate the potential drop across the potentiometer wire

We will use Ohm's law to determine the potential drop (\(V_{wire}\)) across the potentiometer wire length: \[V_{wire} = I \cdot R_{wire}\] Substitute the values of I and \(R_{wire}\): \[V_{wire} = 0.4~A \cdot 4~\Omega\] \[V_{wire} = 1.6~V\]

Calculate the potential gradient of the wire

The potential gradient (\(k\)) is the potential drop per unit length of the wire: \[k = \frac{V_{wire}}{length}\] Substitute the values of \(V_{wire}\) and length: \[k = \frac{1.6~V}{200~cm}\] \[k = 8 \times 10^{-3}~V~cm^{-1}\] The potential gradient of the wire is \(8 \times 10^{-3}~V~cm^{-1}\), which corresponds to answer (A).