Question

Figure shows the internal wiring of a three-range voltmeter whose binding posts are marked +, 2V, 10V and 100V. When the meter is connected to the circuit being measured, one connection is made to the post marked + and the other to the post marked with the desired voltage range. The resistance of the moving coil $R_G$ is $40\Omega$ and a current of 1 mA in the coil causes it to deflect full-scale. Then match the following:

• A. Value of resistance $R_1$ in $k\Omega$
• B. Value of resistance $R_3$ in $k\Omega$
• C. Overall resistance of the meter in 100 V range in $k\Omega$
• D. Overall resistance of the meter in 2 V range in $k\Omega$

Answer 1

a-r, b-s, c-p, d-q

Answer 2

The problem describes a three-range voltmeter and asks to match the calculated values of resistances and overall meter resistances with given options.

Given:

• Resistance of the moving coil, $R_G = 40 \, \Omega$
• Full-scale deflection current, $I_G = 1 \, \text{mA} = 1 \times 10^{-3} \, \text{A}$

A voltmeter is formed by connecting a high resistance (multiplier) in series with a galvanometer. The voltage range $V$ is given by the formula: $V = I_G (R_G + R_{series})$ where $R_{series}$ is the total series resistance connected with the galvanometer coil.

Let's calculate the required values step-by-step:

a. Value of resistance $R_1$ in k$\Omega$ When the meter is used in the 2V range, the total resistance in the circuit is $R_G + R_1$. The full-scale voltage for this range is $V_1 = 2 \, \text{V}$. $V_1 = I_G (R_G + R_1)$ $2 \, \text{V} = (1 \times 10^{-3} \, \text{A}) (40 \, \Omega + R_1)$ $2 / (1 \times 10^{-3}) = 40 + R_1$ $2000 = 40 + R_1$ $R_1 = 2000 - 40 = 1960 \, \Omega$ Converting to k$\Omega$: $R_1 = 1.96 \, \text{k}\Omega$. So, (a) matches with (r).

b. Value of resistance $R_3$ in k$\Omega$ To find $R_3$, we first need to find $R_2$. For the 10V range, the total resistance in the circuit is $R_G + R_1 + R_2$. The full-scale voltage for this range is $V_2 = 10 \, \text{V}$. $V_2 = I_G (R_G + R_1 + R_2)$ $10 \, \text{V} = (1 \times 10^{-3} \, \text{A}) (40 \, \Omega + 1960 \, \Omega + R_2)$ $10 / (1 \times 10^{-3}) = 2000 + R_2$ $10000 = 2000 + R_2$ $R_2 = 10000 - 2000 = 8000 \, \Omega = 8 \, \text{k}\Omega$.

Now, for the 100V range, the total resistance in the circuit is $R_G + R_1 + R_2 + R_3$. The full-scale voltage for this range is $V_3 = 100 \, \text{V}$. $V_3 = I_G (R_G + R_1 + R_2 + R_3)$ $100 \, \text{V} = (1 \times 10^{-3} \, \text{A}) (40 \, \Omega + 1960 \, \Omega + 8000 \, \Omega + R_3)$ $100 / (1 \times 10^{-3}) = 10000 + R_3$ $100000 = 10000 + R_3$ $R_3 = 100000 - 10000 = 90000 \, \Omega$ Converting to k$\Omega$: $R_3 = 90 \, \text{k}\Omega$. So, (b) matches with (s).

c. Overall resistance of the meter in 100 V range in k$\Omega$ The overall resistance of the meter in the 100V range is the total resistance in series with the galvanometer for this range. $R_{total, 100V} = R_G + R_1 + R_2 + R_3$ $R_{total, 100V} = 40 \, \Omega + 1960 \, \Omega + 8000 \, \Omega + 90000 \, \Omega$ $R_{total, 100V} = 100000 \, \Omega$ Converting to k$\Omega$: $R_{total, 100V} = 100 \, \text{k}\Omega$. So, (c) matches with (p).

d. Overall resistance of the meter in 2 V range in k$\Omega$ The overall resistance of the meter in the 2V range is the total resistance in series with the galvanometer for this range. $R_{total, 2V} = R_G + R_1$ $R_{total, 2V} = 40 \, \Omega + 1960 \, \Omega$ $R_{total, 2V} = 2000 \, \Omega$ Converting to k$\Omega$: $R_{total, 2V} = 2 \, \text{k}\Omega$. So, (d) matches with (q).

Summary of Matches:

• a. Value of resistance $R_1$ in k$\Omega$ $\rightarrow$ r. 1.96
• b. Value of resistance $R_3$ in k$\Omega$ $\rightarrow$ s. 90
• c. Overall resistance of the meter in 100 V range in k$\Omega$ $\rightarrow$ p. 100
• d. Overall resistance of the meter in 2 V range in k$\Omega$ $\rightarrow$ q. 2

Explanation of the solution:

1. Voltmeter Principle: A voltmeter measures potential difference by connecting a high resistance in series with a galvanometer. The full-scale deflection voltage ($V$) is related to the full-scale deflection current ($I_G$), galvanometer resistance ($R_G$), and series resistance ($R_{series}$) by $V = I_G (R_G + R_{series})$.
2. 2V Range: For the 2V range, the series resistance is $R_1$. Using $V_1 = I_G (R_G + R_1)$, calculate $R_1$. The overall resistance for this range is $R_G + R_1$.
3. 10V Range: For the 10V range, the series resistance is $R_1 + R_2$. Using $V_2 = I_G (R_G + R_1 + R_2)$, calculate $R_2$.
4. 100V Range: For the 100V range, the series resistance is $R_1 + R_2 + R_3$. Using $V_3 = I_G (R_G + R_1 + R_2 + R_3)$, calculate $R_3$. The overall resistance for this range is $R_G + R_1 + R_2 + R_3$.
5. Conversion: Convert all resistance values from Ohms to kilo-Ohms as required.