Step 1: General formula for converting galvanometer to voltmeter
A galvanometer can be converted into a voltmeter by connecting a resistance \( R \) in series. The relationship is given by:\[V = I_g (R + G)\]where \( V \) is the full-scale deflection voltage, \( I_g \) is the current for full-scale deflection of the galvanometer, \( G \) is the resistance of the galvanometer, \( R \) is the series resistance to be calculated.Case 1: Using resistance \( R_1 \) to measure up to \( V \) \[V = I_g (R_1 + G) \quad \cdots (1)\]Case 2: Using resistance \( R_2 \) to measure up to \( \frac{V}{2} \) \[\frac{V}{2} = I_g (R_2 + G) \quad \cdots (2)\]Dividing (1) by (2): \[\frac{V}{\frac{V}{2}} = \frac{R_1 + G}{R_2 + G} \Rightarrow 2 = \frac{R_1 + G}{R_2 + G}\Rightarrow R_1 + G = 2(R_2 + G)\Rightarrow R_1 + G = 2R_2 + 2G\Rightarrow R_1 - 2R_2 = G\quad \cdots (3)\]Step 2: Find \( R_3 \) for 2 V full-scale deflection
Let \( R_3 \) be the series resistance required to measure 2 V. The equation is:\[2 = I_g (R_3 + G) \quad \cdots (4)\]Substituting \( I_g \) from equation (1):\[I_g = \frac{V}{R_1 + G}\Rightarrow 2 = \frac{V}{R_1 + G} (R_3 + G)\Rightarrow R_3 + G = \frac{2(R_1 + G)}{V}\]Using equation (3): \( G = R_1 - 2R_2 \) \[R_3 + (R_1 - 2R_2) = \frac{2(R_1 + R_1 - 2R_2)}{V}\Rightarrow R_3 = \frac{2(2R_1 - 2R_2)}{V} - (R_1 - 2R_2)\]Simplifying the expression: \[R_3 = \frac{4(R_1 - R_2)}{V} - (R_1 - 2R_2)\]% Final Answer Statement Answer: The required resistance \( R_3 \) is given by:\[\boxed{R_3 = \frac{4(R_1 - R_2)}{V} - (R_1 - 2R_2)}\]