Answer:
The acceleration is ![a = 9197.11 m/s^2](https://tex.z-dn.net/?f=a%20%3D%209197.11%20m%2Fs%5E2)
Explanation:
From the question we are told that
The radius of the cylinder is ![r = 6.5 \ cm = \frac{6.5}{100} = 0.065 m](https://tex.z-dn.net/?f=r%20%3D%206.5%20%5C%20cm%20%3D%20%5Cfrac%7B6.5%7D%7B100%7D%20%3D%200.065%20m)
The magnitude of the magnetic field is ![B_i = 3.5T](https://tex.z-dn.net/?f=B_i%20%3D%203.5T)
The rate of decrease is ![\frac{dB}{dt} = 29.5 \ G/s = \frac{29.5}{10000} = 29.5 * 10 ^{-4} T/s](https://tex.z-dn.net/?f=%5Cfrac%7BdB%7D%7Bdt%7D%20%20%3D%2029.5%20%5C%20G%2Fs%20%3D%20%5Cfrac%7B29.5%7D%7B10000%7D%20%3D%2029.5%20%2A%2010%20%5E%7B-4%7D%20T%2Fs)
The distance from the center is ![D = 1.5\ cm = \frac{1.5}{10}=0.15m](https://tex.z-dn.net/?f=D%20%3D%201.5%5C%20cm%20%3D%20%5Cfrac%7B1.5%7D%7B10%7D%3D0.15m)
Faraday's law of induction states mathematically that
![\epsilon = \frac{d \o}{dt}](https://tex.z-dn.net/?f=%5Cepsilon%20%3D%20%5Cfrac%7Bd%20%5Co%7D%7Bdt%7D)
![\epsilon = \int\limits {E} \, dl](https://tex.z-dn.net/?f=%5Cepsilon%20%3D%20%5Cint%5Climits%20%20%7BE%7D%20%5C%2C%20dl)
Where
is the induced emf
E is the magnitude of the electric field
i the change in length
is the change in magnetic flux which is mathematically represented as
![\o = BA](https://tex.z-dn.net/?f=%5Co%20%3D%20%20BA)
substituting this into the above equation
![E l = A \frac{dB}{dt}](https://tex.z-dn.net/?f=E%20l%20%20%3D%20A%20%5Cfrac%7BdB%7D%7Bdt%7D)
Where l is the circumference of the circular loop formed in the cylinder which is mathematically represented as
![l = 2\pi r](https://tex.z-dn.net/?f=l%20%3D%202%5Cpi%20r)
And A is the area of the circular loop formed which is mathematically represented as
![A = \pi r^2](https://tex.z-dn.net/?f=A%20%3D%20%5Cpi%20r%5E2)
So
![E (2 \pi r ) = (\pi r^2 ) \frac{dB}{dt}](https://tex.z-dn.net/?f=E%20%282%20%5Cpi%20r%20%29%20%3D%20%28%5Cpi%20r%5E2%20%29%20%5Cfrac%7BdB%7D%7Bdt%7D)
![E = \frac{r}{2} \frac{dB}{dt}](https://tex.z-dn.net/?f=E%20%3D%20%5Cfrac%7Br%7D%7B2%7D%20%5Cfrac%7BdB%7D%7Bdt%7D)
Substituting value
![E = \frac{0.065}{2} * 29.5 *10^{-4}](https://tex.z-dn.net/?f=E%20%3D%20%5Cfrac%7B0.065%7D%7B2%7D%20%20%2A%2029.5%20%2A10%5E%7B-4%7D)
![= 9.588*10^{-5} \ V/m](https://tex.z-dn.net/?f=%3D%209.588%2A10%5E%7B-5%7D%20%5C%20V%2Fm)
Generally acceleration is mathematically represented as
![a = \frac{F}{m}](https://tex.z-dn.net/?f=a%20%3D%20%5Cfrac%7BF%7D%7Bm%7D)
Now F is the electric force which is mathematically represented as
![F = qE](https://tex.z-dn.net/?f=F%20%3D%20qE)
Substituting this into the question
![a = \frac{qE }{m }](https://tex.z-dn.net/?f=a%20%3D%20%5Cfrac%7BqE%20%7D%7Bm%20%7D)
Where q is the charge on the proton with a constant value of
![q = 1.60 *10^ {-19 } C](https://tex.z-dn.net/?f=q%20%3D%201.60%20%2A10%5E%20%7B-19%20%7D%20C)
and m is the mass of the proton with a constant value of
Substituting values
![a = \frac{1.60*10^{-19} * 9.588 *10^{-5}}{1.67 *10^{-27}}](https://tex.z-dn.net/?f=a%20%3D%20%5Cfrac%7B1.60%2A10%5E%7B-19%7D%20%2A%209.588%20%2A10%5E%7B-5%7D%7D%7B1.67%20%2A10%5E%7B-27%7D%7D)
![a = 9197.11 m/s^2](https://tex.z-dn.net/?f=a%20%3D%209197.11%20m%2Fs%5E2)