An electron moving to the right at 3.0% the speed of light enters a uniform electric field parallel to its direction of motion. If the electron is to be brought to rest in the space of 4.5 cm, determine the following. What direction is required for the electric field? What is the strength of the field?
This is relatively straightforward. What is your difficulty or question on it?
setting it up
To determine the direction and strength of the electric field required to bring the electron to rest, you can use the concepts of electric force and acceleration.
1. Calculate the initial velocity of the electron:
Since the electron is moving at 3.0% the speed of light, we can calculate its initial velocity as follows:
Speed of light, c = 3.0 x 10^8 m/s
Initial velocity, v = 3.0% of c = 0.03 x c = 0.03 x 3.0 x 10^8 m/s = 9.0 x 10^6 m/s
2. Calculate the acceleration of the electron:
Using the equation for accelerating charged particles in an electric field:
Electric force, F = mass x acceleration
Electric force, F = charge x electric field strength
Since the electron has a negative charge (equal to -1.6 x 10^-19 C), the electric force is directed opposite to the electric field.
Therefore, we have:
F = -e x E (where e is the charge of the electron and E is the electric field strength)
Mass of electron, m = 9.1 x 10^-31 kg
Rearranging the equation, we get:
Acceleration, a = -eE / m
3. Calculate the time required for the electron to come to rest:
When the electron comes to rest, its final velocity will be zero. Using the equation of motion:
Final velocity, Vf = Vi + at
Since Vf = 0, we can rearrange the equation to solve for time:
t = -Vi / a
Inserting the values, we get:
t = (-9.0 x 10^6 m/s) / (-eE / m)
4. Calculate the distance traveled by the electron:
The distance traveled by the electron while coming to rest is given as 4.5 cm. Convert it to meters:
Distance, d = 4.5 cm = 4.5 x 10^-2 m
5. Use the relationship between distance, time, and acceleration:
d = 0.5at^2
Substitute the values of d and t:
4.5 x 10^-2 m = 0.5 x (-eE / m) x [(-9.0 x 10^6 m/s) / (-eE / m)]^2
Simplify the equation by canceling out similar terms.
6. Solve for the unknown electric field strength:
After simplifying, we get:
4.5 x 10^-2 m = (9.0 x 10^6 m^2/s^2) / E
Rearrange the equation to solve for E:
E = (9.0 x 10^6 m^2/s^2) / (4.5 x 10^-2 m)
Once you have solved this equation, you will be able to determine both the direction and strength of the electric field required to bring the electron to rest.