Question

Two particles A and B of masses $m$ and $2m$ have charges $q$ and $2q$ respectively. They are moving with the velocities ${v_1}$ and ${v_2}$ respectively, in the same direction, entering the same magnetic field, $B$ acting normally to their direction of motion. If the two forces ${F_A}$ and ${F_B}$ acting on them are in the ratio of $1:2$, find the ratio of their velocities.

Answer 1

Anything that causes an object to undergo unnatural motion is called force. In other words, the external agent that is capable of changing the object’s state of rest or motion is called a force. To solve the given problem, consider the equation that relates the charge and the velocity.

Formula used:
$\Rightarrow F = qVB$
Where, $F$ is the force, $q$ is the charge, $V$ is the velocity, and $B$ is the magnetic field.

Complete step by step solution:
Given, two particles have masses of $m$ and $2m$. The charges are $q$ and $2q$. The velocities of the bodies are ${v_1}$ and ${v_2}$. The magnetic field $B$ is present.
To find the ratio of the velocities when the forces ${F_A}$ and ${F_B}$ act on them as the ratio of $1:2$.
To solve the given problem the equation of force is used. Force is the external agent that is capable of changing the state of rest or the motion of the objects. The equation of the motion is,
$\Rightarrow F = qVB$
Where, $F$ is the force, $q$ is the charge, $V$ is the velocity, and $B$ is the magnetic field.
There are two particles given. Calculate the force for both the particles separately.
For particle A:
Force acting on particle A is ${F_A} = q{V_1}B$
The value for the charge $q$ is one.
Substitute the values.
$\Rightarrow {F_A} = q{V_1}B$
For particle B:
Force acting on the particle B is ${F_B} = qVB$
The value for the charge $q$ is two.
Substitute the values.
$\Rightarrow {F_B} = 2q{V_2}B$
Divide the force values of A and B.
$\Rightarrow \dfrac{{{F_A}}}{{{F_B}}} = \dfrac{{q{V_1}B}}{{2q{V_2}B}}$
The ratio value of forces is given as $1:2$.
Substitute the value.
$\Rightarrow \dfrac{1}{2} = \dfrac{{q{V_1}B}}{{2q{V_2}B}}$
Cancel out the common terms,
$\Rightarrow \dfrac{1}{2} = \dfrac{{{V_1}}}{{2{V_2}}}$
Simplify the equation.
$\Rightarrow {V_1} = {V_2}$
$\therefore {V_1}:{V_2} = 1:1$
Hence, the ratio of their velocities is $1:1$.

Note:
When a force is applied to a direction, the direction of the force is called the Direction of force, the point to which the force is applied is called the application of the force. The force can be measured by using a spring balance. Newton $\left( N \right)$ is the unit of the force. There are types of forces. Among them, the weakest force is the Gravitational force.