Answer:
The acceleration is [tex]a = 9197.11 m/s^2[/tex]
Explanation:
From the question we are told that
The radius of the cylinder is [tex]r = 6.5 \ cm = \frac{6.5}{100} = 0.065 m[/tex]
The magnitude of the magnetic field is [tex]B_i = 3.5T[/tex]
The rate of decrease is [tex]\frac{dB}{dt} = 29.5 \ G/s = \frac{29.5}{10000} = 29.5 * 10 ^{-4} T/s[/tex]
The distance from the center is [tex]D = 1.5\ cm = \frac{1.5}{10}=0.15m[/tex]
Faraday's law of induction states mathematically that
[tex]\epsilon = \frac{d \o}{dt}[/tex]
[tex]\epsilon = \int\limits {E} \, dl[/tex]
Where [tex]\epsilon[/tex] is the induced emf
E is the magnitude of the electric field
[tex]dl[/tex] i the change in length
[tex]d \o[/tex] is the change in magnetic flux which is mathematically represented as
[tex]\o = BA[/tex]
substituting this into the above equation
[tex]\int\limits {E} \, dl = \frac{d(BA)}{dt }[/tex]
[tex]E l = A \frac{dB}{dt}[/tex]
Where l is the circumference of the circular loop formed in the cylinder which is mathematically represented as
[tex]l = 2\pi r[/tex]
And A is the area of the circular loop formed which is mathematically represented as
[tex]A = \pi r^2[/tex]
So
[tex]E (2 \pi r ) = (\pi r^2 ) \frac{dB}{dt}[/tex]
[tex]E = \frac{r}{2} \frac{dB}{dt}[/tex]
Substituting value
[tex]E = \frac{0.065}{2} * 29.5 *10^{-4}[/tex]
[tex]= 9.588*10^{-5} \ V/m[/tex]
Generally acceleration is mathematically represented as
[tex]a = \frac{F}{m}[/tex]
Now F is the electric force which is mathematically represented as
[tex]F = qE[/tex]
Substituting this into the question
[tex]a = \frac{qE }{m }[/tex]
Where q is the charge on the proton with a constant value of
[tex]q = 1.60 *10^ {-19 } C[/tex]
and m is the mass of the proton with a constant value of
[tex]m = 1.67 *10^{-31} kg[/tex]
Substituting values
[tex]a = \frac{1.60*10^{-19} * 9.588 *10^{-5}}{1.67 *10^{-27}}[/tex]
[tex]a = 9197.11 m/s^2[/tex]
