Question

A meter gauge train is heading north with speed $54km{h^{ - 1}}$ in earth’s magnetic field $3 \times {10^{ - 4}}T$. The $emf$ induced across the axle joining the wheels is
(A) $0.45mV$
(B) $4.5mV$
(C) $45mV$
(D) $450mV$

Answer 1

Hint We are given here with the length of the gauge of the train, the direction and speed of the train and the magnetic field of the earth and are asked to calculate the $emf$ induced across the axle rod joining the wheels. Thus, we will use the formula of emotional $emf$ as this case clearly is assisted by this concept.
Formulae Used
$\varepsilon = BLv$
Where, $\varepsilon $ is the induced$emf$, $B$ is the magnetic field, $L$ is the length of the moving pivot and $v$ is its velocity.

Complete Step by Step Solution
Here,
We can safely assume that the length of the axle is approximately equal to the gauge measurement of the train.
Thus,
$L = 1m$
Now,
Velocity of the axle would be the same as the velocity of the train.
Thus,
$v = 54km{h^{ - 1}}$ due north.
Converting this value into the standard unit, we get
$v = 15m{s^{ - 1}}$
And the magnetic field of the earth is given to be
$B = 3 \times {10^{ - 4}}T$
Thus,
The induced $emf$ will be,
$\varepsilon = (3 \times {10^{ - 4}})(1)(15)$
Which turns out to be,
$\varepsilon = 45 \times {10^{ - 4}}V$
Converting into$mV$, we get
$\varepsilon = 4.5mV$

Hence, the correct option is (B).

Additional information
The concept of emotional $emf$ is just an extension of Faraday's law of electromagnetic induction. It is just the collaboration of the former law with Lenz's law.
The Lenz’s law can be simply put as a conducting material opposes the change in magnetic field attached to it. In other words, if the magnetic field around it is increasing, it tries to decrease the same and vice versa.

Note
Before putting in the values of magnetic field, length and the velocity of the object, we need to make sure that all the three parameters are mutually perpendicular to each other. If any of them are at some other angles to the remaining parameters, then the calculation procedure and the substitution of the formula will be different.