Question

In a TV picture tube electrons are ejected from the cathode with negligible speed and reach a velocity of $5 \times {10^6}\dfrac{m}{s}$ in traveling one centimeter. Assuming straight-line motion, find the constant force exerted on the electron. The mass of the electron is $9.1 \times {10^{ - 31}}kg$.

Answer 1

Hint: When some potential difference is applied across the two ends of a metal conductor, an electric field is set up inside the conductor. Each free-electron experiences a force when it comes to the influence of the electric field. Hence there is an acceleration of the electron.

Complete step by step answer:
Given, initial velocity of the electrons is negligible, that is,$u = 0m{s^{ - 1}}$
Final velocity of the electrons, $v = 5 \times {10^6}m{s^{ - 1}}$
Distance travelled by the electrons, $s = 1cm = 1 \times {10^{ - 2}}m$
From equation of kinematics we have, ${v^2} = {u^2} + 2as$
Above equation can also be written as, acceleration of the electrons, $a = \dfrac{{{v^2} - {u^2}}}{{2s}}$
Substituting the given values we get,
$ \Rightarrow a = \dfrac{{{{\left( {5 \times {{10}^6}} \right)}^2} - {{\left( 0 \right)}^2}}}{{2 \times 1 \times {{10}^{ - 2}}}}$
Simplifying the above equation we get,
$ \Rightarrow a = \dfrac{{25 \times {{10}^{12}}}}{{2 \times 1 \times {{10}^{ - 2}}}}$
$ \Rightarrow a = 12.5 \times {10^{14}}m{s^{ - 2}}$
Then, the constant force exerted on the electron is given by,
From Newton’s law of motion, we have,
$ \Rightarrow F = ma$
Newton’s law of motion states that force is proportional to the products of mass and acceleration.
We have, the mass of the electron is $9.1 \times {10^{ - 31}}kg$.
Substituting the values in the above equation,
$ \Rightarrow F = 9.1 \times {10^{ - 31}} \times 12.5 \times {10^{14}}m{s^{ - 2}}$
$\therefore F = 1.1 \times {10^{ - 15}}N$ .

Hence, the constant force exerted on the electron is $1.1 \times {10^{ - 15}}N$.

Additional information:
The second law of motion provides a method of measuring force. When a force acts on a body it changes the velocity of the body. When the velocity changes, the momentum of the body also changes. If the force is acting in the direction of motion, the momentum increases, and if it is acting in the opposite direction, the momentum decreases.

Note:
Drift velocity is the average velocity at which the free electrons get drifted in the metallic conductor under the influence of the electric field.
Electrons are negatively charged particles.
According to Newton’s second law of motion, change in momentum per unit time is directly proportional to the force acting on it and it takes place in the direction of the force.