The anode (positive terminal) of an X-ray tube is at a potential of +125000 V with respect to the cathode (negative terminal). (a) How much work (in joules) is done by the electric force when an electron is accelerated from the cathode to the anode? (b) If the electron is initially at rest, what kinetic energy does the electron have when it arrives at the anode?
To find the work done by the electric force when an electron is accelerated from the cathode to the anode, we can use the equation:
Work = q * ΔV
where q is the charge of the electron and ΔV is the potential difference between the anode and the cathode.
(a) To calculate the work done, we need to know the charge of the electron. The charge of an electron is approximately -1.6 x 10^(-19) coulombs.
Given:
ΔV = +125000 V (since the anode is at a potential of +125000 V with respect to the cathode)
q = -1.6 x 10^(-19) C
Using the equation for work, we have:
Work = (-1.6 x 10^(-19) C) * (+125000 V)
Calculating this gives us:
Work = -2.0 x 10^(-14) J
Therefore, the work done by the electric force when an electron is accelerated from the cathode to the anode is -2.0 x 10^(-14) joules.
(b) To find the kinetic energy of the electron when it arrives at the anode, we can use the equation for kinetic energy:
Kinetic Energy = (1/2) * m * v^2
where m is the mass of the electron and v is its velocity.
The mass of an electron is approximately 9.11 x 10^(-31) kg.
Given that the electron is initially at rest, its initial velocity (u) is 0 m/s.
We know the work done by the electric force, which can be equated to the change in kinetic energy of the electron. Therefore:
Work = Kinetic Energy - Initial Kinetic Energy
Since the electron is initially at rest, the initial kinetic energy (K.E.) is 0.
Therefore:
-2.0 x 10^(-14) J = Kinetic Energy - 0
Simplifying the equation, we find:
Kinetic Energy = -2.0 x 10^(-14) J
However, kinetic energy cannot be negative for a moving object. Therefore, there is an error, and we need to re-evaluate our approach.
In this case, the given potential difference (ΔV) is the potential energy difference between the cathode and the anode, not the actual kinetic energy of the electron.
To find the kinetic energy of the electron, we can use the conservation of energy principle:
Potential Energy + Kinetic Energy = Total Energy
Initially, the electron has zero kinetic energy. Therefore, the total energy of the electron is equal to its potential energy at the anode.
Potential Energy = q * ΔV
Using the given values:
Potential Energy = (-1.6 x 10^(-19) C) * (+125000 V)
Calculating this gives us:
Potential Energy = -2.0 x 10^(-14) J
Since total energy is the sum of potential and kinetic energy, we can rewrite the equation as:
Total Energy = -2.0 x 10^(-14) J
Now, to find the kinetic energy when the electron arrives at the anode, we rearrange the equation:
Total Energy = Potential Energy + Kinetic Energy
Solving for Kinetic Energy:
Kinetic Energy = Total Energy - Potential Energy
Substituting the values:
Kinetic Energy = 0 J - (-2.0 x 10^(-14) J)
Simplifying, we get:
Kinetic Energy = 2.0 x 10^(-14) J
Therefore, when the electron arrives at the anode, it has a kinetic energy of 2.0 x 10^(-14) joules.