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) 25 A/m² B) 6.2 A/m² C) 12 A/m² D) 17 A/m²

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Quizplus · Accepted Answer

The total current is given by the integral of the current density over the area of the beam:$$I = \int_0^R J(r) 2\pi r dr = \int_0^R J_0 \frac{r}{R} 2\pi r dr = \frac{2\pi J_0 R^2}{2} = \pi J_0 R^2$$Solving for J0, we get:$$J_0 = \frac{I}{\pi R^2} = \frac{1.3 	imes 10^{-3} A}{\pi (5 	imes 10^{-3} m)^2} = 25 A/m^2$$

A Proton Beam That Carries a Total Current of 1