First slide

Resistors

Question

The figure shows a portion of a circuit. Resistor are known and indicated on diagram and voltmeters are identical. If voltmeters $v_{1} a n d v_{2}$ reads 7.5V and 5V respectively then reading of $v_{3}$ .

Moderate

A

0 V
B

10 V
C

2.5 V
D

3.0 V

Solution

[Concept]: Since all voltmeters are identical there resistors have resistance $R_{v}$
$V_{1} = i_{1} R_{v} V_{2} = i_{2} R_{v} V_{3} = i_{3} R_{v}$
$\begin{array}{l} [Step 1]: \frac{i_{1} R_{v}}{i_{2} R_{v}} = \frac{V_{1}}{V_{2}} \Rightarrow \frac{i_{1}}{i_{2}} = \frac{7.5}{5} = \frac{3}{2} \Rightarrow i_{1} = \frac{3}{2} i_{2} \\ \Rightarrow let i_{2} = x a n d i_{1} = \frac{3}{2} x \\ [Step 2]: {loop1 containing V}_{1} and V_{2} : \\ 9 i_{1} - 14 i_{2} + V_{2} - V_{1} = 0 \\ 9 \times \frac{3}{2} x - 14 x + 5 - 7.5 = 0 \\ 13.5 x - 14 x = 2.5 \\ [Step 3] : x = - 5.0 A = I_{2} \\ [Step 4] : i_{1} = \frac{3}{2} i_{2} = \frac{3}{2} (-5)⇒ - 7.5 A = I_{1} \\ V_{1} = i_{1} R_{v} {⇒7.5=7.5R}_{v} \\ {Hence resistance of voltmeter(R}_{v})is 1 Ω \\ [Step 6] : [Step 2]: {loop2 containing V}_{1} and V_{3} : \\ 24 I_{3} = - 7.5 - 7.5 \times 9 = \\ 24 I_{3} = - 75 \\ [Step 7]: I_{3} = - 3 A \\ Hence V_{3} = i_{3} R_{v} =3(1)= 3.0 V \end{array}$

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