**Problem 2**: A 5.0-mm-diameter proton beam carries a total current of \( I = 1.5 \, \text{mA} \). The current density in the proton beam, which increases with distance from the center, is given by \( J = J_{\text{edge}} (r/R) \), where \( R \) is the radius of the beam and \( J_{\text{edge}} \) is the current density at the edge. Determine the value of \( J_{\text{edge}} \). **a)** Fig. 3 shows the cross section of the beam. Compute the current \( dI \) flowing through the ring of radius \( r \) and width \( dr \) shown in the figure. Notice that for small \( dr \) the area of the ring can be approximated by the area of a rectangle that you can get by "unrolling" the ring. **Diagram Explanation**: The diagram illustrates a cross-section of a circular beam with concentric rings. The ring shown has a radius \( r \) and a width \( dr \). By unrolling this annular ring, it approximates a rectangle with width \( dr \) and length \( 2\pi r \). **b)** Sum up the contributions from all rings by integrating \( dI \) with respect to the radial coordinate \( r \), \( I = \int_{r=0}^{r=R} dI \). Express \( J_{\text{edge}} \) as a function of \( I \) and \( R \) and compute its value. **c)** How many protons per second are delivered by this proton beam?

Problem 2: A 5.0-mm-diameter proton beam carries a total current of \( I = 1.5 \, \text{mA} \). The current density in the proton beam, which increases with distance from the center, is given by \( J = J_{\text{edge}} (r/R) \), where \( R \) is the radius of the beam and \( J_{\text{edge}} \) is the current density at the edge. Determine the value of \( J_{\text{edge}} \). a) Fig. 3 shows the cross section of the beam. Compute the current \( dI \) flowing through the ring of radius \( r \) and width \( dr \) shown in the figure. Notice that for small \( dr \) the area of the ring can be approximated by the area of a rectangle that you can get by "unrolling" the ring. Diagram Explanation: The diagram illustrates a cross-section of a circular beam with concentric rings. The ring shown has a radius \( r \) and a width \( dr \). By unrolling this annular ring, it approximates a rectangle with width \( dr \) and length \( 2\pi r \). b) Sum up the contributions from all rings by integrating \( dI \) with respect to the radial coordinate \( r \), \( I = \int_{r=0}^{r=R} dI \). Express \( J_{\text{edge}} \) as a function of \( I \) and \( R \) and compute its value. c) How many protons per second are delivered by this proton beam?