Question

In a region of uniform electric field of intensity E, an electron of mass ${m_e}$is released from rest. The distance travelled by electron in time t is:
A)$\dfrac{{2{m_e}{t^2}}}{e}$
B) $\dfrac{{eE{t^2}}}{{2{m_e}}}$
C) $\dfrac{{g{m_e}{t^2}}}{{eE}}$
D) $\dfrac{{2E{t^2}}}{{e{m_e}}}$

Answer 1

Hint- When a unit positive charge is placed at a point, the electric field of intensity is experienced at that particular point. The electric field intensity is a vector quantity.

Formula used: To solve this type of question we use the following formulas.
$v = u + at$
${v^2} = {u^2} + 2as$
$s = ut + \dfrac{1}{2}a{t^2}$
These are equations of motion. Where u is initial velocity of an object, v is final velocity of the object, a is acceleration and s is the distance.
${F_{E.F.}} = qE$
This is the force acting on charged particles in an electric field. Where q is changed and E is electric field intensity.

Complete step by step answer:
According to the question, electric field intensity is E and the mass of the electron is${m_e}$, which is released in the electric field from rest. That means initial velocity is zero, u=0 and let e be the charge of this electron.
Let us write the force experienced by this electron in electric field using ${F_{E.F.}} = qE$.
${F_{E.F.}} = eE$ (1)
Now we know Newton’s law, which is $F = ma$. So from here we can rewrite equation (1) for electron as follows.
${m_e}a = eE$ (2)
We have to find the distance travelled by the electron in t time. For this we will choose one of the equations of motion. So, let us use $s = ut + \dfrac{1}{2}a{t^2}$.
Let us substitute the values in this equation.
$s = 0 \times t + \dfrac{1}{2}a{t^2}$
Let us further simplify.
$s = \dfrac{1}{2}a{t^2}$
From this equation we will take out the value of acceleration as follows.
 $a = \dfrac{{2s}}{{{t^2}}}$ (3)
Now, let us substitute the value of a from equation (3) to (2).
${m_e}\dfrac{{2s}}{{{t^2}}} = eE$
Let us take out the value of distance ‘s’ from the above expression.
$s = \dfrac{{eE{t^2}}}{{2{m_e}}}$
Hence, option (B) $\dfrac{{eE{t^2}}}{{2{m_e}}}$ is the correct option.

Note: Whenever an object is at rest, its initial velocity u=0.
We can always solve these types of questions using the three equations of motion. However, we have to decide the equation to be used based on the variables given in the question.