A pulsed proton beam is fired at a target. Each pulse lasts 45.0 ns, and there are protons in each pulse, each proton having a speed of m/s. All the protons hit a circular area of , called the beam spot. What is the average pressure on the beam spot during a pulse if all the protons are absorbed by the target

P = n P₀ 4.9 10¹⁴  Pa

Explanation:

The radiation pressure for full absorption is

        P = S / c

Where S the pointing vector, which is equal to the intensity of the beam that is defined as the energy per unit area per unit time

The energy of the protons can be calculated

       Em = K = ½ m v²

Area

          A = π r²

Intensity is

          I = n ½ m v² / π r²

          I = ½ n m /π v² / r²

We replace

         S = U / t A

          S = ½ n m /π v² / r² Δt

The pressure is

           P = 1/c   (½ n m /π (v / r Δt)²2

           Δt = 45 10⁻⁹ s

           P = n [½ m /πc  (v/r)²] 4.9 10¹⁴

The amount in square brackets is the pressure that a proton creates, which is why it is useful

         P = n P₀ 4.9 10¹⁴  Pa

Where Po is the pressure created by a proton