Some forms of cancer can be treated using proton therapy in which proton beams are accelerated to high energies, then directed to collide into a tumor, killing the malignant cells. Suppose a proton accelerator is 4.21 m long and must accelerate protons from rest to a speed of 1.06 x 107 m/s. Ignore any relativistic effects and determine the magnitude of the average electric field that could accelerate these protons.
sqrt(2 a X) = V
Solve the above equation for the required acceleration, a, and then use
a = F/m = eE/m
to solve for the electric field E;
e and m are the proton charge and mass.
To determine the magnitude of the average electric field that could accelerate the protons, we can use the equations of motion for uniformly accelerated motion.
The first step is to find the time it takes for the protons to reach the desired speed. We can use the equation:
v = u + at
Where:
v = final velocity (1.06 x 10^7 m/s)
u = initial velocity (0 m/s)
a = acceleration (unknown)
t = time
Since the protons start from rest, the initial velocity (u) is 0, so the equation becomes:
v = at
Solving for t:
t = v/a
Now, we can find the acceleration (a) using the formula for average acceleration:
a = (v - u) / t
Substituting the values:
a = (1.06 x 10^7 m/s - 0 m/s) / t
Next, we need to find the distance (d) that the protons travel. The length of the proton accelerator is given as 4.21 m. So:
d = 4.21 m
We can use the equation of motion again to relate distance, initial velocity, time, and acceleration:
d = ut + (1/2)at^2
Since the initial velocity is 0, the equation simplifies to:
d = (1/2)at^2
Substituting the values:
4.21 m = (1/2) * a * t^2
We can rearrange this equation to solve for t:
t^2 = (2 * d) / a
t = sqrt((2 * d) / a)
Now, we have both the time (t) and acceleration (a) in terms of the variables we know.
Finally, we can determine the average electric field (E) that could accelerate the protons. We can use the equation:
E = ΔV / d
Where:
E = electric field
ΔV = change in voltage
d = distance
In this case, the change in voltage is equal to the kinetic energy gained by the protons:
ΔV = KE / q
Where:
KE = kinetic energy
q = charge of the proton
The formula for kinetic energy is:
KE = (1/2)mv^2
Where:
m = mass of the proton
v = final velocity
Substituting the values:
KE = (1/2) * (1.67 x 10^-27 kg) * (1.06 x 10^7 m/s)^2
Now, we can calculate the charge of the proton:
q = 1.6 x 10^-19 C
Substituting all the values into the equation for electric field (E):
E = ((1/2) * (1.67 x 10^-27 kg) * (1.06 x 10^7 m/s)^2) / (1.6 x 10^-19 C * 4.21 m)
Calculating this expression will give us the magnitude of the average electric field that could accelerate the protons.