Introduction:
In order to construct a multi-range voltmeter using a galvanometer circuit, we need to determine the appropriate values of resistances R1, R2, and R3. The galvanometer has a resistance of 10 Ohms and produces a maximum deflection of 1 milliampere. We want to design a voltmeter that can measure up to 2 volts, 200 volts, and 20 volts.

Understanding the Galvanometer Circuit:
The galvanometer is a sensitive current measuring device that can be converted into a voltmeter by connecting it in parallel with a resistor. When a current flows through the galvanometer, it creates a magnetic field that deflects the needle. By adding a resistor in parallel, we can divert a portion of the current away from the galvanometer and limit the deflection.

Designing the Voltmeter:
To design the voltmeter, we need to ensure that the maximum current flowing through the galvanometer does not exceed 1 milliampere for any of the voltage ranges.

Range 1 - 2 Volts:
In this range, the maximum current through the galvanometer should be 1 milliampere. Let's assume the resistance R1 is connected in parallel with the galvanometer. Using Ohm's law (V = IR), we can calculate the resistance required to limit the current to 1 milliampere.

I = 1 milliampere = 0.001 Ampere
V = 2 Volts
R1 = V/I = 2/0.001 = 2000 Ohms

Therefore, the value of R1 should be 2000 Ohms.

Range 2 - 200 Volts:
In this range, we need to limit the current to 1 milliampere as well. To achieve this, we can connect a resistor R2 in series with the combination of R1 and the galvanometer. The total resistance should be such that the current remains within 1 milliampere.

I = 1 milliampere = 0.001 Ampere
V = 200 Volts

Using Ohm's law (V = IR), we can calculate the total resistance:
Rtotal = V/I = 200/0.001 = 200000 Ohms

Since Rtotal is the sum of R1, R2, and the galvanometer resistance (10 Ohms), we can calculate R2 as follows:
Rtotal = R1 + R2 + Rgalvanometer
R2 = Rtotal - R1 - Rgalvanometer
R2 = 200000 - 2000 - 10 = 197990 Ohms

Therefore, the value of R2 should be 197990 Ohms.

Range 3 - 20 Volts:
Similar to the previous range, we need to limit the current to 1 milliampere. We can achieve this by connecting a resistor R3 in series with the combination of R1, R2, and the galvanometer. The total resistance should be such that the current remains within 1 milliampere.

I = 1 milliampere = 0.001 Ampere
V = 20 Volts

Using Ohm