Problem 4 - Master Resource Book in JEE Main Physics

Question

The wing span of an aeroplane is \(36 \mathrm{~m}\). If the plane is flying at \(400 \mathrm{kmh}^{-1}\), the emf induced between the wings tips is (Assume \(V=4 \times 10^{-5} \mathrm{~T}\) ) (a) \(16 \mathrm{~V}\) (b) \(1.6 \mathrm{~V}\) (c) \(0.16 \mathrm{~V}\) (d) \(0.016 \mathrm{~V}\)

Step-by-Step Solution

Answer

The emf induced is 1.6 V.

Step 1: Formula for EMF

To find the EMF induced between the wing tips, we use the formula for motional EMF: \( \text{EMF} = B \times l \times v \), where \( B \) is the magnetic field strength, \( l \) is the length of the conductor (span of wings), and \( v \) is the velocity of the plane.

Step 2: Convert Units

First, convert the plane's speed from km/h to m/s for consistency, as \( v = 400 \, \text{km/h} = \frac{400 \times 1000}{3600} \, \text{m/s} \). Simplifying, we find \( v = 111.11 \, \text{m/s} \).

Step 3: Plug in Values

Substitute the given values into the EMF formula: \( B = 4 \times 10^{-5} \, \text{T} \), \( l = 36 \, \text{m} \), \( v = 111.11 \, \text{m/s} \). Calculate the EMF: \( \text{EMF} = 4 \times 10^{-5} \times 36 \times 111.11 \).

Step 4: Calculate EMF

Compute the product: \( \text{EMF} = 4 \times 36 \times 111.11 \times 10^{-5} \). Performing the multiplication, we find \( \text{EMF} = 1.6 \, \text{V} \).

Key Concepts

Electromagnetic Induction

Electromagnetic induction is a fascinating phenomenon where an electromotive force (EMF) is generated by changing magnetic fields. This is a fundamental principle behind many electrical devices like transformers and electric generators. Micheal Faraday, in the 1830s, discovered this principle which opened pathways to modern electrical engineering. Faraday’s Law states that the EMF induced is directly proportional to the rate of change of the magnetic flux. In simpler terms, moving a conductor (like a wire) through a magnetic field can induce an EMF. This is the same concept that is being used in our airplane exercise, where the movement of the plane through Earth's magnetic field induces voltage between the wing tips, setting the stage for understanding practical applications.

Magnetic Field

Magnetic fields are invisible forces that exert a push or pull on certain materials such as iron, and they are pervasive throughout the universe. One of the key characteristics of a magnetic field is its strength and direction, typically represented by more concentrated field lines indicating stronger fields. In the context of the problem, the Earth's magnetic field was assumed at a strength of 4x10^-5 Tesla.
The magnetic field facilitates the induction process by interacting with the moving airplane wings, acting as a crucial medium through which energy conversion takes place. As the conductor moves perpendicular to the magnetic field, an EMF is generated making use of the spatial separation of charges within the sprayed field.

Velocity Conversion

In physics problems, consistent units are crucial to obtaining the correct results. In this exercise, the airplane's speed was originally given in kilometers per hour (km/h), which needed conversion to meters per second (m/s).
To convert velocity from km/h to m/s, you can use the conversion factor: 1 km/h is equal to approximately 0.27778 m/s. For precise conversion, the speed is multiplied by \( \frac{1000}{3600} \) to obtain m/s. This is because there are 1000 meters in a kilometer and 3600 seconds in an hour.
Such conversions are essential because the formula for motional EMF uses metric units: Tesla (T) for magnetic field strength, meters (m) for length, and meters per second (m/s) for velocity, enabling accurate calculation of the induced EMF.