A potentiometer wire of length \(L\) and a resistance \(r\) are connected in series with a battery of EMF \(E_{0 }\) and resistance \(r_{1}\). An unknown EMF is balanced at a length l of the potentiometer wire. The EMF \(E\) will be given by:
1. \(\frac{L E_{0} r}{l r_{1}}\)
2. \(\frac{E_{0} r}{\left(\right. r + r_{1} \left.\right)} \cdot \frac{l}{L}\)
3. \(\frac{E_{0} l}{L}\)
4. \(\frac{L E_{0} r}{\left(\right. r + r_{1} \left.\right) l}\)

A potentiometer wire has a length of \(4​​~\text{m}\) and resistance  \(8~\Omega.\)  The resistance that must be connected in series with the wire and an energy source of emf \(2~\text{V}\), so as to get a potential gradient of \(1~\text{mV}\) per cm on the wire is:
1. \(32~\Omega\)
2. \(40~\Omega\)
3. \(44~\Omega\)
4. \(48~\Omega\)

The figure given below shows a circuit when resistances in the two arms of the meter bridge are \(5~\Omega\) and \(R\), respectively. When the resistance \(R\) is shunted with equal resistance, the new balance point is at \(1.6l_1\). The resistance \(R\) is: 
           
1. \(10~\Omega\)
2. \(15~\Omega\)
3. \(20~\Omega\)
4. \(25~\Omega\)

A potentiometer circuit has been set up for finding the internal resistance of a given cell. The main battery, used across the potentiometer wire, has an emf of \(2.0~\text{V}\) and negligible internal resistance. The potentiometer wire itself is \(4~\text{m}\) long. When the resistance, \(R\), connected across the given cell, has values of (i) infinity (ii) \(9.5\), the 'balancing lengths, on the potentiometer wire, are found to be \(3~\text{m}\) and \(2.85~\text{m}\), respectively. The value of the internal resistance of the cell is (in ohm):
1. \(0.25\)
2. \(0.95\)
3. \(0.5\)
4. \(0.75\)

A potentiometer circuit is set up as shown in the figure below. The potential gradient across the potentiometer wire is k volt/cm. Ammeter present in the circuit reads 1.0 A when the two-way key is switched off. The balance points, when the key between the terminals (i) 1 and 2 (ii) 1 and 3, is plugged in, are found to be at lengths $l_1$ $cm$ and $l_2$ $cm$ respectively. The magnitudes of the resistors R and X in ohm, are then, respectively, equal to: 
   

1. $k(l_2-l_1)$ $and$ $k{l}_2$

2. $k{l}_1$ $and$ $k(l_2-l_1)$

3. $k(l_2-l_1)$ $and$ $k{l}_1$

4. $k{l}_1$ $and$ $k{l}_2$

A cell can be balanced against 100 cm and 110 cm of potentiometer wire, respectively with and without being short-circuited through a resistance of 10 Ω. Its internal resistance is:

1.  1.0 Ω

2.  0.5 Ω

3.  2.0 Ω

4.  zero