Question

One micro-ammeter has a resistance of \(100 \Omega\) and a full scale range of \(50 \mu \mathrm{A}\). ft can be used as a voltmeter or as a higher range ammeter provided a resistance is added to it. Pick the correct range and resistance combinations (s). (A) \(5 \mathrm{~mA}\) range with \(1 \Omega\) resistance in parallel (B) \(10 \mathrm{~mA}\) range with \(1 \Omega\) resistance in parallel (C) \(10 \mathrm{~V}\) range with \(200 \mathrm{k} \Omega\) resistance in series (D) \(50 \mathrm{~V}\) range with \(10 \mathrm{k} \Omega\) resistance in series

Short answer

The correct range and resistance combinations are (C) $10 \mathrm{~V}$ range with $200 \mathrm{k} \Omega$ resistance in series.

Step-by-step solution

Recall Ohm's Law

Let's recall Ohm's Law: V = IR, where V represents voltage, I represents current, and R represents resistance.

Calculate the voltage across the micro-ammeter

We use the given full-scale range of the micro-ammeter (50 μA) and its resistance (100 Ω) to calculate the voltage across it. \[V = IR = (50 \times 10^{-6} A)(100 \Omega)\] \[V = 5 \times 10^{-3} V = 5 mV\]

Analyze each combination

Now we will analyze each combination to determine if it is correct. (A) 5 mA range with 1 Ω resistance in parallel: We want the total current to be 5 mA when the micro-ammeter reads its full-scale value of 50 μA. Total resistance in parallel: \[R_{total} = \frac{R_{meter} \times R_{shunt}}{R_{meter} + R_{shunt}}\] \[R_{total} = \frac{100 \Omega \times 1 \Omega}{100 \Omega + 1 \Omega}\] Now, find the current through the 1 Ω shunt resistor, using the voltage calculated in Step 2 (5 mV). \[I = \frac{V}{R} = \frac{5 \times 10^{-3} V}{R_{total}}\] If this current is close to 5 mA, then this combination is correct. (B) 10 mA range with 1 Ω resistance in parallel: Repeat the process from the previous step, but with a target current of 10 mA. (C) 10 V range with 200 kΩ resistance in series: For this combination, we want the voltage across the micro-ammeter and the additional resistor to be 10 V when the micro-ammeter reads its full-scale value of 50 μA. Calculate the total resistance: \[R_{total} = R_{meter} + R_{additional} = 100 \Omega + 200 \times 10^3 \Omega\] Now, find the voltage across the total resistance, using the full-scale current (50 μA). \[V = IR = (50 \times 10^{-6} A)(R_{total})\] If this voltage is close to 10 V, then this combination is correct. (D) 50 V range with 10 kΩ resistance in series: Repeat the process from the previous step, but with a target voltage of 50 V.

Compare the results

Compare the results from Step 3 to determine which combinations are correct. The correct ones should have a voltage or current close to the target values.

Key concepts

Micro-ammeter

A micro-ammeter is an instrument used to measure small electrical currents, typically in the microampere range. It is highly sensitive and suitable for applications where precise low-current measurements are needed.
The micro-ammeter mentioned in the exercise has a full-scale range of 50 μA and a resistance of 100 Ω. This means it can accurately measure up to 50 μA before reaching full scale.
When transforming a micro-ammeter into a voltmeter or a higher range ammeter, additional resistance needs to be added. This can be achieved by either placing a resistor in parallel or in series with the micro-ammeter, depending on the desired outcome.

• Adding a parallel resistor allows it to measure higher currents.
• Adding a series resistor allows it to measure higher voltages.

Parallel and Series Circuits

Electrical circuits can be connected in two basic ways: parallel and series, each with distinct characteristics and applications. Understanding these configurations is key to modifying devices like the micro-ammeter.
In a parallel circuit, components are connected side by side. Current divides among the branches but the voltage across each component is the same. When a 1 Ω resistance is added in parallel, the micro-ammeter can handle additional current by distributing it between the device and the added resistance.
In a series circuit, components are connected end to end. The current throughout the circuit remains the same, but the total voltage is the sum of the voltages across each component. Resistors added in series increase the device's voltage range.

• Parallel circuits help in increasing current capacity.
• Series circuits help in increasing voltage capacity.

Current Calculation

Accurate current calculations are vital for safely expanding a micro-ammeter's measuring range. Using Ohm's Law, where V = IR, current (I) can be calculated if voltage (V) and resistance (R) are known.
For example, if a parallel setup is desired, and the target is a 5 mA range, the total road map involves:
1. Calculating the total resistance using the formula for parallel resistors.
2. Using the calculated voltage from the micro-ammeter's specifications.
3. Solving for current using I = \[ \frac{V}{R_{total}} \]
By adhering to this method, each potential configuration of resistors can be examined to ensure that it achieves the desired increase in range safely and correctly.

Voltage Measurement

Voltage measurement with a micro-ammeter involves extending its capability beyond current measurement. When configured with series resistors, it acts as a voltmeter by dividing the total voltage across the added resistor and the micro-ammeter.
To calculate the necessary resistance for a given voltage range, like 10 V or 50 V from our original exercise, follow these steps:
1. Determine the required total resistance with the formula R_{total} = R_{meter} + R_{additional}.
2. Calculate the appropriate voltage through the resistor by using the full-scale current rating of the micro-ammeter, utilizing V = IR.
This process allows the micro-ammeter to measure voltages significantly higher than its original range by ensuring that the total voltage aligns with the device’s full-scale current capacity.