A microwave source of frequency 25.0 GHz (where G means \"gigs\", which means 10

Question

A microwave source of frequency 25.0 GHz (where G means "gigs", which means 10^) pulses that last 1.50 ns (where means, " " which means 10^). A parabolic reflector with a circular cross-sectional diameter of 16.0 cm these into a parallel beam. The average power produced by each pulse is 25.0 kW (where k means "kilo, " which means 10^3). (i) What is the total energy in each pulse (in Joules)? (ii) What is the total energy density in each pulse (in Joules/m^3)? (iii) What is the amplitude of the electric field in this pulse (in V/m)? (iv) Mixed nuclear waste straight out of a commercial nuclear reactor has a half-life of 600 years. One kilogram of nuclear waste has an activity of 10^13 Bq. How many years will it take for this waste to decay to the activity that ordinary granite has, which is 10^3 Bq? (v) Suppose that 1 gram of this waste is plutonium 239, which as a half-life of 26, 100 years, and an activity of 6.29 times 10^11 Bq. How many years will it take for this plutonium to decay to the activity that ordinary granite has, which is 10^3 Bq?

Explanation / Answer

As frequency, f = 25 GHz,

Time interval, t = 1.5 ns,

Power, P = 25 kW,

A

Total energy in each pulse, E = (25 kW)(1.5 ns) = 37.5x10-6 J

B

Energy density, u = P/A*c = (25x103)/(200.96x10-4)(3x108) = 0.0041468 W/m3

C

As u = 0.5*0*E02

That means

E0 = 3.06126x104 V/m

D

A = A0 (1/2)t/T

t/T = log (A0/A )/log (2)

t = T*{log (A0/A )/log (2)}

t = (600 years)*33.22 = 19931.57 years

E

A = A0 (1/2)t/T

t/T = log (A0/A )/log (2)

t = T*{log (A0/A )/log (2)}

t = (26100 years)*29.2283.322 = 762863.453 years

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