A thin cylindrical shell of radius R 1=6.4cm is surrounded by a second cylindric

Question

A thin cylindrical shell of radius R1=6.4cm is surrounded by a second cylindrical shell of radius R2=8.8cm as in the figure (Figure 1) . Both cylinders are 12 m long and the inner one carries a total charge Q1=0.85C and the outer one Q2=+1.44C.

A) If an electron (m=9.1×1031kg) escaped from the surface of the inner cylinder with negligible speed, what would be its speed when it reached the outer cylinder? Express your answer using two significant figures.

B) If a proton (m=1.67×1027kg) revolves in a circular orbit of radius r=7.7cm about the axis (i.e., between the cylinders), what must be its speed? Express your answer using two significant figures.

Explanation / Answer

Using Guass Law,

for the space between the two cylindrical surfaces,

E A = qin / e0

E ( 2pi r L) = - 0.85 x 10^-6 / 8.854 x 10^-12

E =-1273.26/r

where 0.064 < r < 0.088 m

F = qE

Work done takin it from r1 to r2 = Intergral of qEdr

W = -1273.26q ln(r2/r1)

W = -1273.26 x -1.6 x 10^-19 x ln(0.088/0.064)

= 6.49 x 10^-17 J

Using work energy theorem.

W = mv^2 /2 - 0

6.49 x 10^-17 = 9.109 x 10^-31 x v^2 /2

v =1.19 x 10^7 m/s

B) at r = 0.077 m

F = qE = 1.6 x 10^-19 x 1273.26/0.077 = 2.65 x 10^-15 N

F = mv^2 / r

2.65 x 10^-15 = 1.673 x 10^-27 x v^2 / 0.077

v = 3.49 x10^5 m/s

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