An electron gun has a uniform electric field of 425 N/C. An electron starts from rest and is fired from the
gun, reaching a speed of 3.25 × 10^5𝑚/𝑠. Using the conservation of energy for electric charges, how far
does the electron move relative to the field?
To determine how far the electron moves relative to the electric field, we first need to understand the concept of electric potential energy and kinetic energy. The conservation of energy states that the sum of an object's potential energy and kinetic energy remains constant, assuming no external forces are acting on the object.
In this case, the initial potential energy of the electron in the electric field is converted into kinetic energy as it accelerates. We can use this principle to calculate the distance the electron moves.
First, let's determine the potential energy of the electron at the starting point. The electric potential energy is given by the formula:
PE = q * V
Where PE is the potential energy, q is the charge of the electron (1.6 × 10^-19 C), and V is the electric potential (which is equal to the electric field strength E times the distance d moved by the electron).
PE = q * E * d ...(1)
Now, let's calculate the kinetic energy of the electron when it reaches the final speed. The kinetic energy formula is given by:
KE = (1/2) * m * v^2
Where KE is the kinetic energy, m is the mass of the electron (9.1 × 10^-31 kg), and v is the final velocity of the electron.
KE = (1/2) * m * v^2 ...(2)
Since energy is conserved, the initial potential energy (PE) at the starting point is equal to the final kinetic energy (KE) at the endpoint:
q * E * d = (1/2) * m * v^2
Now, we can solve for the distance (d) by rearranging the equation:
d = (1/2) * (m * v^2) / (q * E)
Substituting the given values into the equation:
d = (1/2) * [(9.1 × 10^-31 kg) * (3.25 × 10^5 m/s)^2] / [(1.6 × 10^-19 C) * (425 N/C)]
By solving this equation, you can find the distance traveled by the electron relative to the electric field.