Understanding Galvanometer Conversions

A galvanometer is a sensitive instrument used to detect small electric currents. It can be modified to function as either an ammeter (to measure current) or a voltmeter (to measure voltage). The resistance of the resulting instrument depends heavily on how the conversion is done and the desired measurement range.

Ammeter Conversion

To convert a galvanometer into an ammeter, a low-resistance wire, called a shunt resistor ($R_s$), is connected in parallel with the galvanometer coil. This allows most of the current to bypass the galvanometer, protecting it and enabling the measurement of larger currents.

The total resistance ($R_A$) of the ammeter is the equivalent resistance of the parallel combination of the galvanometer resistance ($R_g$) and the shunt resistance ($R_s$): $$ R_A = \frac{R_g \times R_s}{R_g + R_s} $$ Since $R_s$ is chosen to be very small, the total resistance $R_A$ is also very small, typically much less than $R_g$. A higher current range (e.g., 10 A compared to 5 A) requires a larger shunt resistance ($R_s$), but the overall ammeter resistance ($R_A$) remains very low.

Voltmeter Conversion

To convert a galvanometer into a voltmeter, a high-resistance wire, called a multiplier resistor ($R_m$), is connected in series with the galvanometer coil. This increases the total resistance, ensuring that only a small current flows through the galvanometer when a potential difference is applied, thereby measuring voltage accurately without significantly loading the circuit.

The total resistance ($R_V$) of the voltmeter is the sum of the galvanometer resistance ($R_g$) and the multiplier resistance ($R_m$): $$ R_V = R_g + R_m $$ The multiplier resistance $R_m$ is determined by the desired voltage range ($V$) and the galvanometer's current sensitivity ($I_g$). The relationship is $V = I_g (R_g + R_m)$. Thus, $R_m = \frac{V}{I_g} - R_g$. To measure a larger voltage range (e.g., 10 V compared to 5 V), a larger multiplier resistance ($R_m$) is needed. Consequently, the total voltmeter resistance ($R_V$) increases with the voltage range.

Comparing Resistances

Based on the conversion methods:

• Ammeter resistance ($R_A$) is always very low.
• Voltmeter resistance ($R_V$) is always very high.
• Voltmeter resistance ($R_V$) increases as the voltage range increases.

Comparing the given options:

• Options 1 and 3 are ammeters. Their resistances will be very small.
• Options 2 and 4 are voltmeters. Their resistances will be significantly higher than the ammeters.
• Comparing the two voltmeters, the one with the larger voltage range (10 V) will require a larger multiplier resistance ($R_m$) than the one with the smaller range (5 V).

Therefore, the voltmeter with a range of 10 V will have the largest resistance.

Conclusion

The device with the largest resistance is the voltmeter designed to measure the highest voltage range, as it necessitates the inclusion of a high-value series resistor (multiplier).