Problem 64 - Master Resource Book in JEE Main Physics

Question

A galvanometer of resistance \(20 \Omega\) shown deflection of 10 divisions. When a current of \(1 \mathrm{~mA}\) is passed through it. If a shunt of \(4 \Omega\) is connected and there are 50 divisions on the scale, the range of the galvanometer is [Kerala CET 2007] (a) \(1 \mathrm{~A}\) (b) \(3 \mathrm{~A}\) (c) \(10 \mathrm{~mA}\) (d) \(30 \mathrm{~mA}\) (e) \(11 \mathrm{~mA}\)

Step-by-Step Solution

Answer

The range of the galvanometer is 30 mA (option d).

Step 1: Understand the problem

We are given a galvanometer with an initial resistance of \(20 \Omega\) showing a deflection of 10 divisions for a current of \(1 \mathrm{~mA}\). A shunt of \(4 \Omega\) is connected, and now the galvanometer has a total of 50 divisions. We need to find the new current range of the galvanometer.

Step 2: Calculate the full-scale deflection current

Initially, the galvanometer shows 10 divisions for \(1 \mathrm{~mA}\). Therefore, the current for full-scale deflection (50 divisions) before any shunt is added can be calculated as follows:\[I_{fs} = \frac{1 \mathrm{~mA} \times 50}{10} = 5 \mathrm{~mA}\]

Step 3: Calculate total resistance with shunt

The total resistance \(R_t\) of the galvanometer and shunt in parallel is given by:\[R_t = \frac{R_g \times R_s}{R_g + R_s} = \frac{20 \Omega \times 4 \Omega}{20 \Omega + 4 \Omega} = \frac{80}{24} \Omega \approx 3.33 \Omega\]

Step 4: Calculate the total current with new shunt

Using the formula for parallel resistances, the total current \(I\) that can pass through the galvanometer system with the shunt is given by:\[I = I_{fs} \left(1 + \frac{R_g}{R_s}\right) = 5 \mathrm{~mA} \left(1 + \frac{20 \Omega}{4 \Omega}\right) = 5 \mathrm{~mA} \times 6 = 30 \mathrm{~mA}\]

Step 5: Conclusion

The range of the galvanometer with the shunt connected is \(30 \mathrm{~mA}\). Therefore, the correct answer is option (d) \(30 \mathrm{~mA}\).

Key Concepts

Understanding Shunt Resistors

A shunt resistor is a low-resistance connection placed in parallel with a device, like a galvanometer. Its main purpose is to "shunt," or divert, a portion of the current away from the primary path. This allows devices to measure larger currents without being damaged. A galvanometer, for example, can measure tiny currents on its own. With a shunt resistor, it can measure much larger currents.
Key points to remember:

• A shunt resistor lowers the effective overall resistance of the circuit by adding another pathway for the current.
• By diverting the current, a shunt helps a galvanometer handle much higher currents without exceeding its limit.
• The value of the shunt resistor is crucial. It is typically less than the resistance of the galvanometer to ensure the majority of the current is diverted.

This concept is vital in extending the measurement range of sensitive instruments.

Exploring Current Range

The current range is the span of current values a device can measure. It is determined by the design and components used in the circuit, such as resistors and galvanometers. In the given problem, the galvanometer with a shunt allows us to measure a broader range of currents, extending from its initial small current limit to a much larger one.
Important considerations:

• The range is determined before and after modifications like adding a shunt resistor.
• For a galvanometer, the range can be extended by decreasing the effective resistance using a shunt.
• Understanding the limits of your measurement tool is crucial to avoid damage and ensure accurate readings.

By ingeniously adjusting the current range, engineers ensure that devices can operate safely under a variety of conditions.

Parallel Resistance in Galvanometer Circuits

Parallel resistance is a concept where two or more resistors are connected across the same voltage source. In the context of a galvanometer and shunt, this applies where both resistors share the same voltage across them. This connection decreases the overall circuit resistance, allowing more current to flow.
Key aspects:

• The relationship of resistors in parallel is given by the formula: \[R_t = \frac{R_1 \times R_2}{R_1 + R_2}\]where \(R_t\) is the total resistance, and \(R_1\) and \(R_2\) are the individual resistances.
• For a galvanometer, this setup aids in spreading the current, making large currents manageable and measurable safely.
• When resistors are in parallel, the total resistance will always be less than the smallest resistor in the set.

Understanding how parallel resistance works is essential for designing circuits capable of handling high currents without overheating or overloading components.

Full-Scale Deflection: What It Means

Full-scale deflection in a measurement device like a galvanometer refers to the point at which the device's needle or display shows the maximum value. This tells us that the maximum current the device can measure is passing through it. Before a shunt is applied, the galvanometer's full-scale deflection is limited; after a shunt is added, this limit increases.
Key points:

• Full-scale deflection is crucial for determining how much current an instrument can handle before it needs adjusting.
• The initial full-scale deflection without a shunt defines the base measurement capacity of the instrument.
• By adding a shunt, the full-scale capacity increases, allowing it to measure larger currents by diverting excess current away from the device.

This concept helps ensure accuracy and safety when measuring electrical currents.

Ohm’s Law and Its Application

Ohm’s Law is fundamental in understanding and designing electrical circuits. It establishes the relationship between voltage ( V ), current ( I ), and resistance ( R ) through the equation: \[ V = I \times R \]In a galvanometer circuit:

• Ohm's Law helps calculate the necessary resistance values to ensure the device functions correctly.
• It is used to determine the required shunt resistance and overall resistance of the circuit when connected in parallel.
• Ohm's Law aids in predicting how changes in resistance or voltage will affect the current flow in the circuit.

By applying Ohm's Law, engineers can optimize devices to make accurate electrical measurements, enhancing their applicability across different scenarios.