Problem 19 - 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 voltmeter reading is \(10^{-3} \text{ V}\) (option c).

Step 1: Convert speed to SI units

First, we need to convert the speed of the train from kilometers per hour (km/h) to meters per second (m/s) because SI units work best for the calculations. We know that 1 km/h is equal to \( \frac{1}{3.6} \) m/s. Therefore, the speed of the train in meters per second is:\[v = 180 \text{ km/h} = 180 \times \frac{1}{3.6} \text{ m/s} = 50 \text{ m/s}\]

Step 2: Identify relevant formula

The problem involves the motion of a conductor (the train and rails) through a magnetic field, creating an electromotive force (emf). The relevant formula for the emf generated is:\[ \text{emf} = B \cdot l \cdot v \]where \(B\) is the magnetic field (vertical component), \(l\) is the separation between the rails, and \(v\) is the speed of the train.

Step 3: Substitute values into the formula

Now substitute the given values into the formula:\[ B = 0.2 \times 10^{-4} \text{ Wb/m}^2, \quad l = 1 \text{ m}, \quad v = 50 \text{ m/s} \]So the emf (or the reading of the voltmeter) is:\[\text{emf} = (0.2 \times 10^{-4}) \cdot 1 \cdot 50 = 10^{-3} \text{ V}\]

Step 4: Choose the correct answer

Based on our calculation in Step 3, the emf (or voltmeter reading) is \(10^{-3} \text{ V}\). This matches option (c) in the given options.

Key Concepts

Magnetic Field

The magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. It is characterized by both a direction and magnitude. In the context of electromagnetic induction, the magnetic field can cause a force which acts on moving charges.

In our exercise, we specifically focus on the vertical component of the Earth's magnetic field, which is a natural magnetic field surrounding the Earth. It is crucial to consider this component because it intersects with the railway track, which in this case, acts as a conductor. When a magnetic field interacts with a conductor, and motion is involved, it can induce an electromotive force (emf). This interaction is central to understanding how electromagnetic induction works.

Key elements include:

• The magnetic field's strength, which is given as the vertical component in units of Weber per square meter (Wb/m²). In our case, it is \(0.2 \times 10^{-4}\text{ Wb/m}^2\).
• The direction of the magnetic field, which in this problem is vertical.

Understanding these aspects of the magnetic field helps explain how the emf is generated as the train moves along the track.

Electromotive Force (emf)

Electromotive force, often abbreviated as emf, is a fundamental concept in electromagnetism. It represents the voltage created by a source of electrical energy, such as an electric generator or a battery. Rather than being a "force" in the traditional sense, it is actually a potential difference, expressed in volts, between two points.

In our scenario, a train moving along a railway track essentially acts as a generator when the magnetic field interacts with the conducting rails. The separation of the rails and their conductivity allows them to generate an emf as the train moves.

The generation of emf in this context is given by the formula:\[\text{emf} = B \cdot l \cdot v\]where:

• \(B\) is the magnetic field strength.
• \(l\) is the separation between the rails (1 meter).
• \(v\) is the velocity of the train (in meters per second).

Using these values, the calculation of emf allows us to determine the voltage observed in the voltmeter connected across the rails. It provides insight into how motion through a magnetic field can translate into electrical energy, a principle harnessed in many technologies.

Railway Track

A railway track is a system consisting of two parallel rails that guide and support a train. In our exercise, the railway tracks are not only serving their typical routing function but also operate as conductors in an electromagnetic induction scenario.

Here, the separation between the two rails is essential for calculating their role in generating an electromotive force (emf). In our problem, this distance is given as 1 meter. This configuration allows the vertical component of the Earth's magnetic field to interact with the rails effectively, inducing an emf as the train moves.

Railway tracks make an excellent example of practical electromagnetic applications:

• They help conduct and measure the emf directly due to their material properties and parallel alignment.
• The induction generated not only contributes to theoretical learning but also to real-world applications like signaling systems and energy measuring.

In exploring such a typical railway track scenario, we gain a deeper understanding of how theories of electromagnetism translate into everyday practicalities.