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 4.0 m long and must accelerate protons from rest to a speed of 1.0 × 107 m/s. Ignore any relativistic effects (Chapter 26) and determine the magnitude of the average electric field that could accelerate these protons.
F = q E = m a = m (Vf/t)
but t = distance/average velocity
and average velocity = Vf/2
so t = 2d/Vf
so
q E = m Vf (Vf/2d) = m Vf^2/2d
E = m Vf^2 /(2qd)
To determine the magnitude of the average electric field required to accelerate the protons, we can use the following formula:
E = ΔV / d
Where:
E is the electric field strength
ΔV is the change in voltage (acceleration potential)
d is the distance over which the voltage is applied (accelerator length)
Given:
ΔV = velocity_final - velocity_initial = (1.0 × 10^7 m/s) - 0 = 1.0 × 10^7 m/s
d = 4.0 m
Now we can calculate the electric field strength:
E = (1.0 × 10^7 m/s) / (4.0 m)
E ≈ 2.5 × 10^6 V/m
Therefore, the magnitude of the average electric field that could accelerate these protons is approximately 2.5 × 10^6 V/m.