Electron in a Constant Electric Field !
An animation illustrating the curved path taken by an electron in a constant electric field. Revealing the speed and direction of the electron.
An electron flies into a homogenous electric field against the direction of the line of force. The initial electron velocity/speed is 10 km s-1. The electric field intensity is 20 V m-1. Calculate the speed/velocity of the electron when is has already moved/traveled 9 cm in the field.
Comment: This task can be solved by two different ways:
I. analysing electron movement
II. using the law of conservation of energy
What force (think about it’s size) affects the electron? What is the direction of that force?
What kind of movement does the electron undergo?
Accelerattion of the electron can be figured out by applying Newton’s second law of motion.
A moving electron is affected by an electrical force proportional to the homogenous field intensity. The charge of an electron is negative, therefore the affecting force has the opposite direction in comparation with the electric field vector. So the force will be accelerating the electron.
To evaluate the velocity of the electron, we can use the uniformly accelerated motion trajectory length formula (we have to not forget initial velocity). The acceleration can be evaluated using Newton’s second law and the time can be evaluated using the uniformly accelerated motion velocity formula.
The electron is affected by the force, which size is given by
Electron charge is negative therefore the direction of the force affecting the electron is oposite to the direction of the electric intensity field E⃗ E→. So the force will uniformly accelerate the electron. Using Newton’s second law of motion we can evaluate the acceleration a:
The electric force can be substituted from formula
The electron moves with uniform acceleration, so its velocity is
(v0 is the initial velocity of the electron)
We can evaluate the time t from that equation.
The lenght of the trajectory of uniformly accelerated movement is
Now we can use the evaluation of time from
And evaluate the velocity v from this formula.
Now we can substitute the acceleration a from formula to obtain the electron’s final velocity.
Let’s use the conservation of energy law to calculate electron’s velocity.
The electron is affected by the electric force. It accelerates the electron and so the electron’s kinetic energy is increasing.
Note: The work done by the electric force is equal to the change in the potential energy.