1. The problem statement, all variables and given/known data
A beam of positively charged particles with radius ##R## (current ##I##) is shot from a region of potential ##-V## through a potential difference of ##+V## to a grounded plate. The beam passes through a slit on this plate then passes through another region of length ##L## meters and passes through a slit on a second plate before hitting a target. The proton density is ##p## protons per cubic meter.
a. What is the speed, ##v## of the particles when the beam reaches the first slit?
b. What is the electric field as a function of the radius within and outside the beam?
c. What is the electric potential as a function of the radius if the potential at the center of the beam is ##0##?
d. Due to particle interactions, some of beam diverges over the distance ##L##. If the radial field is constant and ##1/2## the maximum field, estimate how far a proton would move radially due to this field over the distance ##L##.
2. Relevant equations
##U=qV##
##K=\frac{1}{2}mv^2##
##\rho = \frac{Q}{Volume}##
##E=\frac{Q}{4\pi ε_0 R^2}##
## V =\frac{Q}{4 \pi ε_0 R}##
3. The attempt at a solution
a. The initial electric potential energy of the particles is converted to kinetic energy as the particles reach the first plate, so ##qΔV=\frac{1}{2}mv^2 \rightarrow v = \sqrt{\frac{2qV}{m}}##
b. The volume of the beam over the distance ##L## is ##V= (\pi)(R^2)(L)## so there are ##p\pi R^2 L## protons and the total charge is## (1.6*10^{-19})p\pi R^2 L = Q##. So the electric field within the beam at a distance ##r<R## is ##E=\frac{Qr}{4\pi ε_0 R^3}## and for ##r>R##, ##E=\frac{Q}{4\pi ε_0 r^2}##
c.For ##r<R##, the electric potential is ##V=\frac{Q}{4\pi ε_0 R}## and for ##r>R, V=\frac{Q}{4\pi ε_0 r}##
d. No idea
A beam of positively charged particles with radius ##R## (current ##I##) is shot from a region of potential ##-V## through a potential difference of ##+V## to a grounded plate. The beam passes through a slit on this plate then passes through another region of length ##L## meters and passes through a slit on a second plate before hitting a target. The proton density is ##p## protons per cubic meter.
a. What is the speed, ##v## of the particles when the beam reaches the first slit?
b. What is the electric field as a function of the radius within and outside the beam?
c. What is the electric potential as a function of the radius if the potential at the center of the beam is ##0##?
d. Due to particle interactions, some of beam diverges over the distance ##L##. If the radial field is constant and ##1/2## the maximum field, estimate how far a proton would move radially due to this field over the distance ##L##.
2. Relevant equations
##U=qV##
##K=\frac{1}{2}mv^2##
##\rho = \frac{Q}{Volume}##
##E=\frac{Q}{4\pi ε_0 R^2}##
## V =\frac{Q}{4 \pi ε_0 R}##
3. The attempt at a solution
a. The initial electric potential energy of the particles is converted to kinetic energy as the particles reach the first plate, so ##qΔV=\frac{1}{2}mv^2 \rightarrow v = \sqrt{\frac{2qV}{m}}##
b. The volume of the beam over the distance ##L## is ##V= (\pi)(R^2)(L)## so there are ##p\pi R^2 L## protons and the total charge is## (1.6*10^{-19})p\pi R^2 L = Q##. So the electric field within the beam at a distance ##r<R## is ##E=\frac{Qr}{4\pi ε_0 R^3}## and for ##r>R##, ##E=\frac{Q}{4\pi ε_0 r^2}##
c.For ##r<R##, the electric potential is ##V=\frac{Q}{4\pi ε_0 R}## and for ##r>R, V=\frac{Q}{4\pi ε_0 r}##
d. No idea
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