Question 51: Current Electricity

Question

Two identical moving coil galvanometers have $$10 \Omega$$ resistance and full-scale deflection at $$2 \mu \mathrm{A}$$ current. One of them is converted into a voltmeter of range $$10 \mathrm{~mV}$$ and the other into an ammeter of range $$1 \mathrm{~mA}$$ using appropriate resistors. The ratio of resistance of the converted voltmeter to that of the ammeter is

Accepted Answer

To convert the galvanometer into a voltmeter, we need to find the additional series resistance required. The full-scale deflection voltage for the galvanometer itself is given by:
$$ V_g = I_g 	imes R_g = 2 	imes 10^{-6} \, 	ext{A} 	imes 10 \, \Omega = 2 	imes 10^{-5} \, 	ext{V} $$
To achieve a full-scale deflection of 10 mV, the total resistance $R_v$ of the voltmeter should satisfy:
$$ 10 	imes 10^{-3} \, 	ext{V} = I_g 	imes (R_g + R_s) $$
Where $R_s$ is the series resistance. Solving for $R_s$, we have:
$$ R_s = \frac{10 	imes 10^{-3}}{2 	imes 10^{-6}} - 10 = 5000 - 10 = 4990 \, \Omega $$
Thus, the resistance of the voltmeter is:
$$ R_v = R_g + R_s = 10 + 4990 = 5000 \, \Omega $$
To convert the galvanometer into an ammeter of range 1 mA, we need to find the shunt resistance $R_a$. For the galvanometer to show full-scale deflection at 1 mA, the remaining current (beyond the 2 ÃÂµA for full deflection) should pass through the shunt:
$$ I_{shunt} = 1 \, 	ext{mA} - 2 \, \mu 	ext{A} = 0.998 \, 	ext{mA} $$
The voltage across the galvanometer (and hence the shunt) is:
$$ V = I_g 	imes R_g = 2 	imes 10^{-6} \, 	ext{A} 	imes 10 \, \Omega = 2 	imes 10^{-5} \, 	ext{V} $$
Then $R_a$, the shunt resistance, can be calculated using:
$$ R_a = \frac{V}{I_{shunt}} = \frac{2 	imes 10^{-5}}{0.998 	imes 10^{-3}} \approx 0.02004 \, \Omega $$
Finally, to find the ratio of the resistance of the converted voltmeter to that of the ammeter:
$$ 	ext{Ratio} = \frac{R_v}{R_a} = \frac{5000}{0.02004} \approx 249800 \approx 250000 $$
Therefore, the ratio is closest to:
Option C: 250000 : 1

Answer

5000 : 1

Answer

2500 : 1

Answer

250000 : 1

Answer

25000 : 1

Current Electricity

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