The incorrect option in the given circuit if the reading of ammeter is zero is

A. The value of ${\varepsilon _1}$ will be $\dfrac{{\varepsilon (R + r)}}{R}$
B. Current in R is $\dfrac{{{\varepsilon _1}}}{{R + r}}$
C. Value of ${\varepsilon _1}$will be $\varepsilon $
D. Potential across 2R is zero.

Hint: The question is from the current electricity section of physics. We need to have basic knowledge of concepts like resistance, current, potential difference to solve this problem.

Formula used:
Equation of the current is given as,
$I = \dfrac{V}{R}$
Where, $V$ is the potential difference and $R$ is the resistance.

Complete step by step solution:
The ammeter reading is zero, so the current will only flow in the upper circuit. The circuit diagram is shown below.

Equation of the current, $I = \dfrac{V}{R}$.
Here the current will be,
$i = \dfrac{{{\varepsilon _1}}}{{R + r}} = {i_r} = {i_R}$
The potential difference between the points:
${V_A} - {V_D} = {V_B} - {V_E} = {V_C} - {V_F} = \varepsilon $.....(1)
Also the potential difference between the points A and D:
${V_A} - {V_D} = {\varepsilon _1} - r{i_r}$
Substituting the value or ${i_r}$.
${V_A} - {V_D} = {\varepsilon _1} - r\dfrac{{{\varepsilon _1}}}{{R + r}} \\ $
$\Rightarrow {V_A} - {V_D} = \dfrac{{{\varepsilon _1}R}}{{R + r}} \\$
$\Rightarrow \dfrac{{{\varepsilon _1}R}}{{R + r}} = \varepsilon \\ $........from eq.(1)
$\Rightarrow {\varepsilon _1} = \dfrac{{\varepsilon (R + r)}}{R} \\ $
Option (A) is correct. The value of ${\varepsilon _1}$ is $\dfrac{{\varepsilon (R + r)}}{R}$
Option (B) is correct. Current in R (${i_R}$) is $\dfrac{{{\varepsilon _1}}}{{R + r}}$
Option (D) is correct. Potential across 2R is zero. Because no current is flowing through 2R.
Option (C) is incorrect. Value of ${\varepsilon _1} \ne \varepsilon $

Hence, the correct option is option C.

Additional Information: The difference in charge carriers' energy between two places in a circuit is known as the potential difference. Due to the potential difference between the two ends of the battery, current will flow through a wire if its two ends are connected to the opposite ends of the same battery.

Resistance blocks the flow of current. The S.I. unit of resistance is ohms. The current decreases as resistance increases. On the other hand, the current increases as the resistance decreases. A conductor's electrical resistance is affected by the following parameters: The conductor's cross-sectional area, the conductor's length, the conductor's material and the conducting material's temperature. Electrical resistance is inversely proportional to the cross-sectional area and directly proportional to the conductor's length.

Note: In this type of problem, most of the students make mistakes in taking the direction of flow of current through the battery. Always that current must flow from positive terminal to negative terminal if the battery.