An electron moving to the right at 4.0% 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 5.0 cm, determine what the strength of the field is.
At only 4% of the speed of light, you can neglect relativity effects and assume the electron kinetic energy starts out as
(1/2)mV^2 = (1/1250)m c^2
where I used V = c/25.
m is the electron mass.
To stop the electron in a distance X = 0.05 m,
E*X = (1/1250)mc^2
Solve for E, the required field strength.
To determine the strength of the electric field, we can use the formula for the force experienced by a charged particle in an electric field.
The force experienced by a charged particle in an electric field is given by the equation:
F = qE
where F is the force, q is the charge of the particle, and E is the strength of the electric field.
In this case, we know that the electron is moving to the right at 4.0% the speed of light. The speed of light, c, is approximately 3.0 x 10^8 meters per second, so the velocity of the electron (v) can be calculated as:
v = 0.04c
Next, we need to calculate the initial kinetic energy of the electron (K), which is given by the equation:
K = (1/2)mv^2
where m is the mass of the electron and v is its velocity. The mass of an electron, m, is approximately 9.11 x 10^-31 kilograms.
Now, assuming that the electron is brought to rest by the electric field alone, the work done by the electric field (W) is equal to the initial kinetic energy of the electron:
W = K
The work done by the electric field is given by the equation:
W = qEd
where q is the charge of the electron, E is the strength of the electric field, and d is the distance over which the electron is brought to rest.
We are given that the distance (d) is 5.0 cm, which can be converted to meters as:
d = 5.0 x 10^-2 meters
Therefore, we can substitute the known values into the equation:
qEd = (1/2)mv^2
Solving for E, we get:
E = (1/2)mv^2 / (qd)
Substituting the values of m, v, q, and d, we can calculate the strength of the electric field (E).