Question

The correct expression for Lorentz force is
(A) $q\left[ {\vec E + \left( {\vec B \times \vec V} \right)} \right]$
(B) $q\left[ {\vec E + \left( {\vec V \times \vec B} \right)} \right]$
(C) $q\left( {\vec V \times \vec B} \right)$
(D) $q\vec E$

Answer 1

Hint : To solve this question we have to use the definition of the Lorentz force. Then using that definition, we can deduce its expression in the form of the electric and the magnetic fields.

Formula used: The formulae which are used to solve this question are given by
$\Rightarrow {\vec F_E} = q\vec E$ , here ${\vec F_E}$ is the electric force acting on a charge $q$ due to an electric field $\vec E$ .
$\Rightarrow {\vec F_B} = q\left( {\vec V \times \vec B} \right)$ , here ${\vec F_B}$ is the magnetic force acting on a charge $q$ moving with a velocity $\vec V$ in a region of the magnetic field $\vec B$ .

Complete step by step answer
We know that the Lorentz force is the electromagnetic force acting on a charged particle which moves in an electromagnetic field. We know that the electromagnetic field is composed of two field components. One is the electric field, while the other is the magnetic field. So both these fields apply force on the charged particle. Hence, Lorentz force is equal to the sum of the electric and the magnetic forces.
Now, we know that the force on a charged particle due to an electric field is given by
$\Rightarrow {\vec F_E} = q\vec E$ …………………………………..(1)
Also, the force due to a magnetic field on a moving charge is given by
$\Rightarrow {\vec F_B} = q\left( {\vec V \times \vec B} \right)$ ………………….(2)
According to the definition, the Lorentz force is equal to the vector sum of the electric and the magnetic forces. So we have the Lorentz force as
$\Rightarrow \vec F = {\vec F_E} + {\vec F_B}$
From (1) and (2) we get
$\Rightarrow \vec F = q\left[ {\vec E + \left( {\vec V \times \vec B} \right)} \right]$
Hence, the correct answer is option B.

Note
The Lorentz force on a charged particle is observed in many engineering devices. These include the cyclotrons, the magnetrons, mass spectrometers etc.