The perfect formula used for calculating induced emf in a rod moving in a uniform magnetic field is:
A) \[e = \overrightarrow B .(\overrightarrow l \times \overrightarrow v )\]
B) $e = \overrightarrow B .(\overrightarrow l .\overrightarrow v )$
C) $e = \overrightarrow B \times (\overrightarrow l .\overrightarrow v )$
D) $e = \overrightarrow B \times (\overrightarrow l \times \overrightarrow v )$

Hint: Recall that when the magnetic flux changes in a coil then the potential difference is produced. This potential difference is known as electromotive force or emf. It is said to be induced when flux in the coil or the conductor changes.

Complete solution:
When a straight rod will be placed in the magnetic field, then the induced emf will be produced across its end due to the changing magnetic flux.
Let the magnetic flux density across the rod is ‘B’
The length of the conductor be ‘l’
The conductor moves with a velocity ‘v’
Let $\theta $ be the angle between the magnetic field and the direction of motion of the rod
The amount of flux crossing through the rod will be equal to the amount of induced emf produced in the rod. Let the induced emf produced in the straight rod is = e.
Therefore it can be written that
$ \Rightarrow e = Blv\sin \theta $
In the vector form, the above equation can be written as
$ \Rightarrow e = \overrightarrow B .(\overrightarrow l \times \overrightarrow v )$

Option A is the right answer.

Note: It is already clear that the changing magnetic flux produces an induced emf. This process is also known as electromagnetic induction. But it is important to remember that the emf can be induced in different ways. If the area of the loop is changed, then the induced emf is produced. Also if the angle between the magnetic field and the closed loop or rod is changed, then also induced emf can be produced in the rod.