Question

The electron beam in a T.V picture tube travels a distance, l in evacuated space of the tube. If the speed of the electrons is v and the beam current is i. What is the number of electrons in a beam at any instant?
A) $\dfrac{{il}}{{ev}}$
B) $\dfrac{i}{{lev}}$
C) $\dfrac{v}{{ile}}$
D) $\dfrac{e}{{ilv}}$

Answer 1

Use the expression of current,
$i = \dfrac{{n \times e}}{t}$
From this expression find out the expression for number of electrons
Then use the relation of time, velocity and distance to find the expression for time.

Complete step by step answer:
The flow of charge in time t is called Current.
From the definition of current,
$i = \dfrac{Q}{t} \cdots \left( i \right)$
The charge comprises of electrons so, $Q = n \times e$
where, n is the total number of electrons each having charge e.
In equation $\left( i \right)$, $i = \dfrac{{ne}}{t}$
$\therefore n = \dfrac{{i \times t}}{e} \cdots \left( {ii} \right)$
We know that, the relation between speed, distance and time is
$s = \dfrac{d}{t}$(where, s = speed, d = distance and t = time)
The speed of electrons in the electron beam during the distance travelled $l$in evacuated space of tube is
$v = \dfrac{l}{t} \\\ \therefore t = \dfrac{l}{v} \\\$
Using this equation $\left( {ii} \right)$, we get
$n = \dfrac{{i \times \left( {\dfrac{l}{v}} \right)}}{e} \\\ \Rightarrow n = \dfrac{{il}}{{ev}} \\\$
So, the option (A) is correct.

Note: The current is the flow of charge in a time across a cross-sectional area. The charge consists of electrons generally.
So, Charge = $n \times e$
The relation between Distance, Speed and Time Taken is
$D = S \times t$
Where, D = Distance, S = Speed, t = Time taken
Also note that the direction of current is opposite to the direction of movement of electrons.