Question

A galvanometer, having a resistance of \(50 \Omega\), gives a full scale deflection for a current of \(0.05 \mathrm{~A}\). The length in meter of a resistance wire of area of cross-section \(2.97 \times 10^{2} \mathrm{~cm}^{2}\) that can be used to convert the galvanometer into an ammeter which can read a maximum of \(5 \mathrm{~A}\) current is (specific resistance of wire \(=5 \times 10^{-7} \Omega \mathrm{m}\) ) (A) \(1.5\) (B) 6 (C) 8 (D) 3

Short answer

The length of the resistance wire required to convert the galvanometer into an ammeter that can read a maximum current of 5 A is approximately 6 meters. Therefore, the correct answer is choice (B).

Step-by-step solution

Calculate the required resistance to convert the galvanometer into an ammeter

Given that the galvanometer has a resistance of 50 Ω and a full-scale deflection current of 0.05 A. We need to convert it into an ammeter that can read a maximum current of 5 A. Let's call the required resistance R_shunt which will be connected in parallel with the galvanometer. Using the formula for resistors in parallel: \[ \frac{1}{R_T} = \frac{1}{R_G} + \frac{1}{R_{shunt}} \] where, R_T = total resistance of the parallel combination R_G = resistance of the galvanometer = 50 Ω R_shunt = required resistance to convert the galvanometer To find R_T, we can use the formula for maximum current: I = V / R_T Since the full-scale deflection current in the galvanometer is 0.05 A, when connected in parallel with the required resistor, the total current passing through the combination will be 5 A. Thus, \[ R_T = \frac{V}{I} = \frac{V}{5} \] where V is the voltage across the galvanometer, which can be found using Ohm's law: V = I * R_G = 0.05 * 50 = 2.5 V Substitute the value of V into the equation for R_T: R_T = 2.5 / 5 = 0.5 Ω Now, we can solve for R_shunt using the formula for resistors in parallel: \[ \frac{1}{0.5} = \frac{1}{50} + \frac{1}{R_{shunt}} \] Solving for R_shunt, we get: \[ R_{shunt} = \frac{1}{\frac{1}{0.5} - \frac{1}{50}} = 1.03 \Omega \]

Find the length of the wire using specific resistance, area of cross-section, and resistance

We are given the specific resistance (ρ) of the wire as \(5 \times 10^{-7} \Omega m\) and the area of cross-section (A) as \(2.97 \times 10^{-2} cm^2 = 2.97 \times 10^{-6} m^2\). We can now use these values and the value of R_shunt to find the length (L) of the wire. Using the formula for resistance, we get: \[ R_{shunt} = \rho \cdot \frac{L}{A} \] Rearrange the formula to find the length: \[ L = \frac{R_{shunt} \cdot A}{\rho} \] Substitute the values into the equation: \[ L = \frac{1.03 \times 2.97 \times 10^{-6}}{5 \times 10^{-7}} \] Calculate the length: \[ L \approx 6.04 m \] So, the length of the resistance wire required to convert the galvanometer into an ammeter that can read a maximum current of 5 A is approximately 6 meters. Therefore, the correct answer is choice (B).

Key concepts

Ohm's Law

Ohm's Law is a fundamental concept in electronics that defines the relationship between voltage, current, and resistance in an electrical circuit. The formula is expressed as \( V = I \times R \), where \( V \) represents the voltage across the resistor, \( I \) is the current flowing through it, and \( R \) is the resistance.
When applying Ohm’s Law, it’s essential to understand each component:

• Voltage (V): Measured in volts (V), it is the potential difference across the resistive component.
• Current (I): Measured in amperes (A), it is the flow of electric charge through the component.
• Resistance (R): Measured in ohms (Ω), it is a measure of how much the component resists the flow of electric current.

Ohm's Law is widely used to analyze electric circuits, helping determine required parameters when others are known. For instance, in the given problem, we used Ohm’s Law to calculate the voltage across the galvanometer. This understanding was crucial in deriving the total resistance required for converting it into an ammeter.

Specific Resistance

Specific resistance, or resistivity, is a property of materials that measures how strongly a given material opposes the flow of electric current. It is symbolized by \( \rho \) (rho) and is expressed in ohm-meters (Ωm).
The formula for specific resistance is:\[ R = \rho \cdot \frac{L}{A} \]where \( R \) is resistance, \( L \) is length, and \( A \) is the cross-sectional area. Specific resistance tells us how good a material is at conducting electricity.
Materials with low specific resistance, such as copper, are excellent conductors, while materials with high specific resistance, like rubber, are insulators.
In the exercise, specific resistance was used to calculate the length of wire needed to form the requisite shunt resistance in parallel with the galvanometer.

Resistor in Parallel

Resistors connected in parallel share the same voltage across them, and their total or "equivalent" resistance decreases when more resistors are added. The formula for calculating the equivalent resistance \( R_T \) of resistors in parallel is:\[\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}\]This formula shows that the total resistance \( R_T \) is always less than the smallest individual resistance in the parallel combination.
In the context of the galvanometer to ammeter conversion, the shunt resistor was connected in parallel with the galvanometer. This configuration allows the galvanometer to be protected from high currents, while allowing the measured current to be shared between the galvanometer and the shunt resistor. The exercise guided us in finding the appropriate shunt resistor to manage the required total current capability.

Ammeter Design

An ammeter is designed to measure current by incorporating a galvanometer with a low resistance shunt wire. The shunt allows most of the current to bypass the sensitive galvanometer, protecting it from high currents.
The essential elements of an ammeter design include:

• Galvanometer: A sensitive instrument capable of detecting small electric currents.
• Shunt Resistor: A low-resistance component connected in parallel to divert excessive current.

The galvanometer measures only a small portion, such as 1/100th of the current, while the vast majority flows through the shunt. This extends the range of currents the ammeter can measure without damaging the galvanometer.
In the exercise, the need to adjust the ammeter's range by using a specific shunt resistance highlighted how the design allows precise current measurement up to higher levels, like 5 A, without loss of accuracy or damage to the sensitive components.