Problem 18 - Master Resource Book in JEE Main Physics

Question

The rails of a railway track insulated from each other and the ground are connected to a millivoltmeter. Find the reading of voltmeter, when a train travels with a speed of \(180 \mathrm{~km} / \mathrm{h}\) along the track. Given that the vertical component of earth magnetic field is \(0.2 \times 10^{-4} \mathrm{~Wb} / \mathrm{m}^{2}\) and the rails are separated by \(1 \mathrm{~m}\) (a) \(10^{-4} \mathrm{~V}\) (b) \(10^{-2} \mathrm{~V}\) (c) \(10^{-3} \mathrm{~V}\) (d) \(1 \mathrm{~V}\)

Step-by-Step Solution

Answer

The reading of the voltmeter is \(10^{-3} \text{ V}\), option (c).

Step 1: Convert Speed to Meters per Second

The train travels at a speed of \(180 \text{ km/h}\). To convert this speed into meters per second (m/s), we use the conversion factor \(1 \text{ km/h} = \frac{5}{18} \text{ m/s}\). Therefore, \(180 \times \frac{5}{18} = 50 \text{ m/s}\).

Step 2: Calculate Electromagnetic Induced Voltage

The formula for the electromotive force (emf) induced in this scenario, where a conductor (the train) moves perpendicular to a magnetic field, is \( ext{emf} = B \times v \times L \), where \(B\) is the magnetic field, \(v\) is the velocity, and \(L\) is the length between the rails. Using the given values: \( \text{emf} = 0.2 \times 10^{-4} \mathrm{~Wb/m}^2 \times 50 \text{ m/s} \times 1 \text{ m} \).

Step 3: Compute the Voltmeter Reading

Plugging in the values: \( ext{emf} = 0.2 \times 10^{-4} \times 50 \). Calculating this gives \( ext{emf} = 10^{-3} \; ext{V} \).

Step 4: Select the Correct Answer

Based on the calculated induced voltage of \(10^{-3} \text{ V}\), we match this result to one of the multiple-choice options. This matches with option (c) \(10^{-3} \text{ V}\).

Key Concepts

Electromotive Force (EMF)

Electromotive Force or EMF is a core concept in electromagnetic induction. It refers to the voltage generated by a changing magnetic field. To understand EMF, imagine it as the reason current flows in a closed circuit. In simple terms, EMF is the electric potential produced by either a changing magnetic field or a moving conductor within a magnetic field.

For a conductor moving through a static magnetic field, the formula for calculating EMF is given by:

• \[ \text{EMF} = B \times v \times L \]
• \(B\) is the magnetic field strength in Weber per square meter (Wb/m²),
• \(v\) is the velocity of the conductor in meters per second (m/s),
• \(L\) is the length of the conductor in meters (m).

When a train moves along the rails, it acts as a conductor between the railway tracks, which "cuts" across the Earth's magnetic field, thus inducing voltage. Understanding how this happens helps grasp how generators and other electrical devices work.

Magnetic Field

A magnetic field is an invisible force field that exerts influence on charged particles and magnetic materials. It is most commonly represented using field lines that depict the direction and strength of the field. Field lines emerge from the North pole of a magnet and enter at the South pole.

Earth itself has a magnetic field due to the molten iron in its core, which is similar to a giant bar magnet. The Earth's magnetic field has both a horizontal and vertical component. In the given problem, the vertical component of Earth's magnetic field is what interacts with the moving train to generate EMF.

• The magnetic field (\( B \)) is measured in units of Weber per square meter (Wb/m²), indicating the flux per unit area.
• In everyday applications, the magnetic field is crucial for functioning devices such as compasses, electric motors, and magnetic data storage.

The study of magnetic fields teaches us about the forces that govern much of the natural and engineered world around us.

Railway Physics

Railway Physics involves the study of forces and dynamics involved in railway systems. In this scenario, railway physics examines how the movement of the train affects and interacts with Earth's magnetic field.

As the train moves on the tracks, an important factor is the distance between the rails, which in this case is 1 meter. This specific distance is relevant because it determines the length of the conductor "cutting" through the magnetic field, as described in the EMF formula.

• This principle is vital for understanding how trains can be used to measure magnetic fields and currents without using traditional power sources.
• The concept of induced voltage or EMF in railway physics is not only fascinating but also practically significant, influencing the design and operation of electric trains and railway safety systems.

Incorporating magnetic field interactions allows for more efficient train design and operation, contributing to innovations in transportation technology.