A proton beam that carries a total current of 1.3 mA has 10.0 mm diameter. The c
ID: 2286626 • Letter: A
Question
A proton beam that carries a total current of 1.3 mA has 10.0 mm diameter. The current density in the proton beam increases linearly with distance from the center. This is expressed mathematically as J(r) = J0 (r/R), where R is the radius of the beam and J0 is the current density at the edge. Determine the value of J0.
A proton beam that carries a total current of 1.3 mA has 10.0 mm diameter. The current density in the proton beam increases linearly with distance from the center. This is expressed mathematically as J(r) = J0 (r/R), where R is the radius of the beam and J0 is the current density at the edge. Determine the value of J0.
Explanation / Answer
The charge on a proton is 1.6*10^-19 C. The no of protons per second is 1.3*10^-3 / 1.6*10^-19
The total current is ?I(r)*dA (the current density integrated over the cross-sectional area). dA = 2*?*r*dr (the area of an annulus of depth dr and radius r). Therefore
I = ?J(r)*2*?*r*dr [r = 0 to R]
I = ?J(0)*(r/R)*2*?*r*dr
I = J(edge)*2*?*?(r