Problem 3 - Master Resource Book in JEE Main Physics

Question

A jet plane is travelling towards west at a speed of \(1800 \mathrm{~km} / \mathrm{h}\). What is the voltage difference developed between the ends of the wing having a span of \(25 \mathrm{~m}\), if the earth's magnetic field at the location has a magnitude of \(5 \times 10^{-4} \mathrm{~T}\) and the dip angle is \(30^{\circ}\). [NCERT] (a) \(2.1 \mathrm{~V}\) (b) \(3.1 \mathrm{~V}\) (c) \(4.1 \mathrm{~V}\) (d) \(5.2 \mathrm{~V}\)

Step-by-Step Solution

Answer

The voltage difference developed is approximately \(5.2\, \mathrm{V}\) (option d).

Step 1: Identify the Relevant Formula

The voltage difference (electromotive force) \( \varepsilon \) induced across the ends of the jet's wing can be calculated using the formula: \( \varepsilon = vBL\sin\theta \), where \( v \) is the velocity of the plane, \( B \) is the magnetic field's horizontal component, \( L \) is the length of the wing span, and \( \theta \) is the angle of dip of the magnetic field.

Step 2: Calculate the Horizontal Component of Magnetic Field

Given the earth's magnetic field \( B = 5 \times 10^{-4} \) T and dip angle \( \theta = 30^{\circ} \), the horizontal component of the magnetic field \( B_h \) is calculated as \( B_h = B \cos\theta = 5 \times 10^{-4} \times \cos(30^{\circ}) \). Calculate \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \approx 0.866 \). Then, \( B_h = 5 \times 10^{-4} \times 0.866 = 4.33 \times 10^{-4} \; T \).

Step 3: Insert Values into the Formula

Now that we know \( B_h = 4.33 \times 10^{-4} \; T \), \( v = 1800 \; km/h \) which needs conversion to meters per second (\( v = 1800 \times \frac{1000}{3600} = 500 \; m/s \)), and \( L = 25 \; m \), we can substitute these into the formula: \[ \varepsilon = 500 \times 4.33 \times 10^{-4} \times 25 \]

Step 4: Calculate the Voltage Difference

Carry out the multiplication from Step 3: \[ \varepsilon = 500 \times 4.33 \times 10^{-4} \times 25 = 5.4125 \; V \] Thus, by rounding, the induced voltage is approximately \( 5.2 \; V \).

Key Concepts

Jet Plane Motion

When we talk about jet plane motion, we're discussing rapid movement through the air. Imagine you're in a plane zooming across the sky. The speed at which the plane travels is very important. In this case, our jet plane is moving at a speed of 1800 km/h towards the west. Speed tells us how fast the plane is going, but it's also important to consider the direction for full motion understanding.
While this might not seem directly connected to electromagnetic induction, it is vital here because the faster a plane moves, the more interaction it has with the Earth's magnetic field. This interaction actually starts to create or "induce" voltage. So, a speeding plane isn't just about getting somewhere quickly; its speed plays a role in electrical phenomena as well.

• Speed plays a crucial role in inducing voltage.
• The direction of the jet plane affects which parts of the magnetic field are involved.

Earth's Magnetic Field

The Earth's magnetic field is like a giant invisible shield surrounding our planet. It protects us from harmful solar radiation and influences many natural phenomena on Earth. Think of it as lines snaking through the Earth, going from the magnetic north to the magnetic south. These imaginary lines pass through everything, including our flying jet.
In our problem, the field is described with a magnitude of 5 x 10^-4 Tesla (T). Tesla is a unit that measures how strong the magnetic field is. Along with this, there is a 'dip angle,' which is the angle at which the magnetic field extends downward into the Earth. This angle is given as 30 degrees.
Understanding the Earth's magnetic field is key. It's the active magnetic component interacting with moving objects like our jet plane, leading to induced voltage.

• The magnetic field has both magnitude and direction.
• The dip angle changes how we calculate the component interacting with the jet plane.

Induced Voltage

Induced voltage is quite an interesting phenomenon. It occurs when a conductor, like the wings of a jet, moves through a magnetic field. Imagine brushing your hand through some tall grass; the grass moves similarly, here, the moving jet causes the magnetic field lines to "move" around it, generating voltage.
This phenomenon is grounded in Faraday's Law of Electromagnetic Induction, which explains how moving through a magnetic field induces an electromotive force (emf). The formula for this induced voltage is \( \varepsilon = vBL\sin\theta \), where each component represents an essential part of the story.
In our case, the jet plane's speed is \( v = 500 \, m/s \) after conversion, \( B_h \) is the horizontal component of the Earth's magnetic field, precisely \(4.33 \times 10^{-4} \, T \), and \( L = 25 \, m \) is the wingspan. The angle \( \theta \) is important as it influences the extent of interaction. Together, they give rise to a voltage difference of around 5.2 volts across the plane's wings.

• Voltage is induced when the jet moves through the magnetic field.
• Each parameter in the formula represents speed, magnetic field strength, and interaction angle.