Animated Solution for Physics - Electromagnetic Induction: An aeroplane with its wings spread $10\text{ m}$, is flying at a speed of $180\text{ km/h}$ in a horizontal direction. The total intensity of Earth's field at that part is $2.5\times 10^{-4}{\text{ Wb/m}}^2$ and the angle of dip is $60^{\circ }$. The emf induced between the tips of the plane wings will be

Motional EMF in the Sky

Imagine an airplane soaring horizontally through the sky. While it cuts through the air to generate lift, it is simultaneously slicing through something invisible: Earth's magnetic field. Because the airplane's wings are made of conducting metal, this motion turns the entire aircraft into a giant flying battery! This phenomenon is a classic example of Motional EMF.

To understand how much voltage is generated across the wingtips, we first need to break down the physical parameters given in the problem:

Wingspan ($l$): $10\text{ m}$ Velocity ($v$): $180\text{ km/h}$ Earth's Magnetic Field ($B$): $2.5\times 10^{-4}\text{ T}$ Angle of Dip ($\theta$): $60^{\circ }$

Earth's magnetic field is not perfectly horizontal; it dips downwards at an angle $\theta$. We can resolve this field into two components: a horizontal component ${\vec{B}}_H=B\cos \theta$ and a vertical component ${\vec{B}}_V=B\sin \theta$.

For an EMF to be induced, the conductor must cut across the magnetic field lines. Mathematically, the motional EMF is given by the scalar triple product $\epsilon =(\vec{v}\times \vec{B})\cdot \vec{l}$.

Since the airplane is flying horizontally, its velocity $\vec{v}$ and its wings $\vec{l}$ both lie in the horizontal plane. The horizontal component of the magnetic field ${\vec{B}}_H$ also lies in this plane. Therefore, the wings do not "cut" the horizontal field lines in a way that produces an EMF along the wingspan.

However, the vertical component ${\vec{B}}_V$ points straight down, perfectly perpendicular to both the velocity and the wings. This is the component that does the heavy lifting! Our master equation simplifies to:

Before we substitute our values, we must avoid a classic physics trap: unit mismatch. The velocity is given in $\text{km/h}$, which we must convert to standard SI units ($\text{m/s}$):

Now, we substitute everything into our EMF equation:

Knowing that $\sin 60^{\circ }=\frac{\sqrt{3}}{2}$, we can simplify the expression:

To match the options provided, we format our answer into millivolts ($\text{mV}$) by shifting the decimal point:

And there we have it! The airplane generates an induced EMF of $108.25\text{ mV}$ across its wingtips. It's a beautiful demonstration of how the abstract laws of electromagnetism operate on a massive scale right above our heads.