Question

An airplane with a wingspan 50 m is flying horizontally with a speed of 360 km/hr over a place where the vertical component of the earth's magnetic field is $2 \times 10^{-4} Wb/m^2$. The potential difference between the tips of the wings would be
A. 0.1 V
B. 1.0 V
C. 0.2 V
D. 0.01 V

Answer 1

Hint:The problem is based on motional emf in electromagnetic induction. If a coil our conductor moves in a magnetic field an emf will be generated across its ends and the current will flow if the circuit is closed.
Formula used: For a conductor translating in a uniform magnetic field of magnitude B, emf produced is given as:
$e = Blv \sin \theta$

Complete step-by-step solution:
When a conductor will move or translate in a uniform magnetic field, due to its velocity it will experience an effective change in its length with respect to time which will affect the magnetic flux through it and changing flux of magnetic field is known to produce emf.
Since our plane is given to be moving horizontally without making any specific angle, we can take $\theta = 90^{\circ}$ which is equal to 1 so we get the formula for flux as:
$\phi = BA$
when the magnetic field and the cross section are perpendicular to each other. We consider that cross section area is $A = lx$.
As B is constant, by Faraday's law we have the magnitude of emf generated written as:
$e = \dfrac{d \phi}{dt}$
which in our case becomes:
$e = \dfrac{d (Blx)}{dt}$
Now length of the plane is constant and B is constant but as it is moving in space its velocity is just
$v = \dfrac{dx}{dt}$
Therefore our formula becomes
$e = Blv$ .
On substituting the given values B = $2 \times 10^{-4}$ Wb/$m^2$ , v = 360 km/hr = 100 m/s and l = 50 m, we get:
$e = 2 \times 10^{-4} \times 100 \times 50 $= 1V .
Therefore the correct answer is option (B).

Note: It is necessary to change the unit of velocity from km/hr to m/s. To do this, velocity should be multiplied by 1000 and divided by 3600. Also, Wb/$m^2$ is the unit of magnetic flux which is equivalent to 1T. The emf induced will always oppose the motion so there will be a negative sign in the formula.