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**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?

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Hello, I got part A,part B and part wrong I was wondering if you can help me with part A,PART B and. Part C and can. you label which one is which and can you do the steps, step by steps so I can understand it thank you

**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?
Transcribed Image Text:**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?
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