Calc Based Physics Energy Of A System?
In the neck of the picture tube of a certain black-and-white television set, an electron gun contains two charged metallic plates 2.80 cm apart. An electric force accelerates each elec- tron in the beam from rest to 9.60% of the speed of light over this distance. (a) Determine the kinetic energy of the electron as it leaves the electron gun. Electrons carry this en- ergy to a phosphorescent material on the inner surface of the television screen, making it glow. For an electron passing between the plates in the electron gun, determine (b) the magnitude of the constant electric force acting on the elec- tron, (c) the acceleration, and (d) the time of flight.
m0 = mass of the electron = 9.10938188x10^(-31) kg
c = speed of light = 3x10^8 m/s
Since we are dealing with a velocity about 1/10 the speed of light, we should see if Newton provides a reasonable approximation or if we have to consider relativistic effects.
Newton:
KE = (1/2)m*v^2
KE = 3.77784x10(-16) J
Einstein: The energy imparted to the electron can be found using:
E = m*c^2 - E0^2
m = m0*/SQRT[1 - (v/c)^2]
E0 = m0*c^2
E = m0*c^2/SQRT[1 - (v/c)^2] - m0*c^2
E = m0*c^2*{1/SQRT[1 - (v/c)^2] - 1}
E = 3.80416x10^(-16) J
So Netwon is off by less than 1 percent so we can use Newtonian methods and be reasonably accurate.
If we consider the KE to be work done by the electric field then:
W = F*d where d = distance between plates = 2.8 cm = 0.028 m
F = W/d = 1.35863x10^(-14) N
To get an acceleration I will use F = m*a which, although not true when realtivistic effects, must be considerd, is fine to use under these conditions.
a = F/m = 1.49146x10^16 m/s^2
Since we are using constant acceleration, the average velocity will just be the final velocity divided by 2. The time to travel between the plates is then the distance divided by this average velocity.
Vavg = Vf/2 = 0.048*c
Time = distance/Vavg = 1.94444x10^(-09) s
Check.
v = a*t = 29000606 m/s = 0.0967 the speed of light. close enough
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