Problem 56 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A galvanometer has an internal resistance of \(32 \Omega\) and deflects full scale for a \(55-\mu\) A current. Describe how to use this galvanometer to make \((a)\) an ammeter to read currents up to \(25 \mathrm{~A}\), and \((b)\) a voltmeter to give a full scale deflection $$\text { of } 250 \mathrm{~V} \text { . }$$

Step-by-Step Solution

Answer

Use a shunt resistor of \(0.0000704 \Omega\) for the ammeter and a series resistor of \(4,545,090 \Omega\) for the voltmeter.

Step 1: Understanding the Problem

We need to convert a galvanometer into an ammeter and a voltmeter. The galvanometer has an internal resistance of \(32 \Omega\) and a full-scale deflection with \(55 \mu A\). We want to make an ammeter for \(25 \text{ A}\) and a voltmeter for \(250 \text{ V}\).

Step 2: Making the Ammeter - Calculating Shunt Resistance

To convert the galvanometer into an ammeter, we use a parallel 'shunt' resistor \(R_s\). The total current is \(25 \text{ A}\), and the galvanometer current is \(55 \mu A\). So, the shunt current \(I_s = 25 - 0.000055\). Applying Ohm's Law: \(V = IR\), we set the voltage across the galvanometer (\(V_g = I_g \times R_g\)) equal to the voltage across the shunt (\(V_s = I_s \times R_s\)). This gives us:\[R_s = \frac{V_g}{I_s} = \frac{0.000055 \times 32}{25 - 0.000055} \approx 0.0000704 \Omega\]

Step 3: Making the Voltmeter - Calculating Series Resistance

For the voltmeter, we add a series resistance \(R_{series}\) to limit the current to \(55 \mu A\) when \(250 \text{ V}\) is applied. Therefore,\[R_{series} = \frac{V - V_g}{I_g} = \frac{250 - (0.000055 \times 32)}{0.000055} \approx 4,545,090 \Omega\]

Key Concepts

Ammeter

An ammeter is a device used to measure the current flowing through a circuit. In order to transform a galvanometer into an ammeter, we need to extend its range, allowing it to measure higher currents. This is achieved by employing a technique known as 'shunting.' Here, we place a low-value 'shunt resistor' in parallel with the galvanometer.

This shunt resistor is essential because it provides an alternative path for the excess current to flow, protecting the delicate galvanometer from high currents. When converting the galvanometer to measure up to 25 A, we calculated the shunt resistance needed to make this conversion effective. In this scenario, the shunt current was nearly the total current, leaving the galvanometer to only handle the small calibration current of 55 µA.

• The parallel resistances ensure the voltage across the galvanometer and shunt are the same.
• Ohm's Law (V = IR) is used to find the appropriate shunt resistance value.
• The shunt resistance value was calculated to be approximately 0.0000704 ohms, allowing most of the 25 A to bypass the galvanometer.

Voltmeter

A voltmeter is used to measure the potential difference, or voltage, across two points in a circuit. In order to utilize a galvanometer as a voltmeter, we must adjust it to handle higher voltages safely. This adjustment is done by adding a high-value series resistor.

The principle behind this is that the series resistor controls the current, ensuring that the galvanometer does not experience more than its full-scale deflection current of 55 µA, even when a higher voltage is applied. In our example, with a required full scale reading of 250 V, we calculated the necessary series resistance.

• Series resistance is added to increase the effective resistance, keeping the current flow within safe limits for the galvanometer.
• The calculated series resistance value was approximately 4,545,090 ohms.
• This ensures the galvanometer handles up to 250 V without exceeding its max current.

Shunt Resistance

Shunt resistance is a small-value resistor connected in parallel with the galvanometer, used in converting a galvanometer into an ammeter. It allows the device to measure larger currents than it could on its own. Here's how it works:

The shunt resistor provides an alternate path for the current, reducing the load on the galvanometer. This division ensures the galvanometer only measures a small portion of total current, while the majority passes through the shunt.

• The voltage across both the galvanometer and shunt must remain equal due to their parallel configuration.
• Calculating the correct shunt resistance involves using Ohm's Law to ensure adequate current distribution.
• In our example, the proposed shunt resistance value of 0.0000704 ohms was needed for a 25 A ammeter conversion.

Understanding shunt resistance is vital for making accurate current measurements with a galvanometer.

Series Resistance

Series resistance plays a crucial role when converting a galvanometer into a voltmeter. By adding a high-value resistor in series with the galvanometer, we can control how much current flows through it when measuring voltage.

This is important because galvanometers are delicate instruments meant for low current use. When used as a voltmeter, the series resistance limits the current, protecting the device from damage while ensuring accurate readings for higher voltages.

• It raises the total resistance in the circuit, thus limiting current to safe levels for the galvanometer.
• In our example, series resistance ensured that the galvanometer could measure up to 250 V with its 55 µA full-scale deflection.
• Calculating series resistance precisely is key to extending the galvanometer's voltage measurement capacity to desired levels.

Correctly calculated series resistance prolongs the lifespan of the galvanometer and enhances its versatility as a voltmeter.