INST 240

RevolutionsLecture 11

Nuclear Energy

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The spacetime momentum vector

• Total length: mc• Length in time direction: γmc• Length in space direction: γmv = γ p

•

•

Pspace

Ptime

mc

γmv

γmc

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Non-relativistic limit• At low velocities (v << c):

“Classical” Physics:

E = ½mv2

p = mv

Relativistic Physics

E = γmc2

p = γmv→ mv + small corrections

→ mc2 + ½m v2+ small corrections

Conserved Energy

• Etotal relativistic= Ptime c = mc2 + ½m v2

• The second term is the kinetic energy!

• We know that in non-relativistic processes it is (often) conserved

• But here, the conserved energy has an additional term that is left even when v0

Erest= mc2

That’s it!

• So Einstein’s famous formula Erest= mc2

turns out to be the energy of an object measured by an observer at rest with respect to the object

• If the object is at rest, it does not have kinetic energy (duh!), but a moving observer will not agree

Agreement

• All observers agree on the length of the energymomentum vector (mc)

• All observers agree that the total relativistic energy is conserved

Etotal relativistic= mc2 + ½m v2 = constant

• Observers will disagree why it is conserved!

Space momentum & Energy divided by c form a vector which

has Einstein’s blessing

• Momentum is conserved, energy is conserved

• Units of momentum are units of energy divided by c

Space&time and Energy&momentum diagrams

time

Momentumspace

Energy

Spacetime and Energymomentum diagrams

c time

Momentumspace

Energy/c

Should energy and momentum have bigger or smaller values in a

frame moving wrt the object?

• A: Both have bigger values

• B: Both have smaller values

• C: Energy stays the same

• D: Momentum stays the same

• E: energy gets bigger, momentum smaller

• F: momentum gets bigger, energy gets smaller

Implications

• “Objects have an intrinsic energy equivalent

to their rest mass”

• “Energy is equivalent to mass”

• E = mc2

(Physicists actually don’t use this form of

the equation, but it’s catchy)

Einstein Original Work (1906)• “Das Prinzip von der Erhaltung der

Schwerpunktsenergie und die Trägheit der Energie” (Only 7 pages!)– “Schreibt man also jeglicher Energie die träge Masse

E/V2 zu, so gilt (…) das Prinzip von der Erhaltung der Bewegung des Schwerpunkts auch auch für Systeme in denen electromagnetische Prozesse vorkommen.”

– With träge Masse as the mass and V=c, we have m= E/c2

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Implications

• Objects have mass.

• Objects that store extra energy have extra mass.

• Objects that have given up all their energy have less mass.

Which has more mass? A hot or a cold Potatoe?

• A: Same

• B: Hot one

• C: Cold one

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Implications

A little bit heavier than... ... all the bits after the explosion

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Implications

A little bit heavier than... ... all the bits after the explosion

total mass = mass of component bits + mass due to “available energy”

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How much heavier?

Trinitrotoluene (TNT)

Take a 1 kg block of TNT.

How much heavier is it than it’s component parts?

Worksheet #5

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How much heavier?

Trinitrotoluene (TNT)

TNT releases 0.65 Calories of energy per gram.

energy in Joules mass in kg

c = 3x108 m/sE = mc2

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How much heavier? - Worksheet

TNT releases 0.65 Calories of energy per gram.E = mc2

1 kg = 1000 g1 Calorie = 4200 Joules

1 kg TNT releases E =2730 kJ = mc2 m = 2,730,000 J/ (300,000,000 m/s)2

= 2.73/9 x 10^6 x 10^-16 kg = 3 x 10^-11 kg

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How much heavier?

1.000000000050 kg

1.000000000000 kg

Note that it has the same number of atoms!The mass comes from the bonds between the atoms

Why should we care?

• Can understand energy production in Sun, stars

• Can produce power by harvesting energy stored in mass, binding energy

• Can construct powerful bombs, too, unfortunately

How do we know how much energy the Sun produces each second?

• The Sun’s energy spreads out in all directions

• We can measure how much energy we receive on Earth

• At a distance of 1 A.U., each square meter receives 1400 Watts of power (the solar constant)

• Multiply by surface of sphere of radius 149.6 bill. meter (=1 A.U.) to obtain total power output of the Sun

Energy Output of the Sun• Total power output: 4 1026 Watts• The same as

– 100 billion 1 megaton nuclear bombs per second

– 4 trillion trillion 100 W light bulbs– $10 quintillion (10 billion billion) worth of

energy per second @ 9¢/kWh

• The source of virtually all our energy (fossil fuels, wind, waterfalls, …)– Exceptions: nuclear power, geothermal

Where does the Energy come from?

• Anaxagoras (500-428 BC): Sun a large hot rock – No, it would cool down too fast

• Combustion?– No, it could last a few thousand years

• 19th Century – gravitational contraction?– No! Even though the lifetime of sun would be

about 100 million years, geological evidence showed that Earth was much older than this

What process can produce so much power?

• For the longest time we did not know

• Only in the 1930’s had science advanced to the point where we could answer this question

• Needed to develop very advanced physics: quantum mechanics and nuclear physics

• Virtually the only process that can do it is nuclear fusion

Nuclear Fusion

• Atoms: electrons orbiting nuclei• Chemistry deals only with

electron orbits (electron exchange glues atoms together to from molecules)

• Nuclear power comes from the nucleus

• Nuclei are very small– If electrons would orbit the

statehouse on I-270, the nucleus would be a soccer ball in Gov. Kasic’s office

– Nuclei: made out of protons (el. positive) and neutrons (neutral)

The Structure of MatterAtom: Nucleus and Electrons

Nucleus: Protons and Neutrons (Nucleons)

Nucleon: 3 Quarks

| 10-10m |

| 10-14m |

|10-15m|

Basic Nuclear Physics• The strong force between protons

and neutrons is short ranged – think velcro!

– Each particle “sticks” only to its neighbors

• The electrical repulsion between protons is weaker but long ranged

– Each proton repels every other one

• Bigger nuclei have more trouble holding together – repulsion eventually wins!

Nuclear fusion reaction– In essence, 4 hydrogen nuclei combine (fuse) to

form a helium nucleus, plus some byproducts (actually, a total of 6 nuclei are involved)

– Mass of products is less than the original mass– The missing mass is emitted in the form of energy,

according to Einstein’s famous formula:

E = mc2

(the speed of light is very large, so there is a lot of energy in even a tiny mass)

• Small nuclei (like hydrogen) can “fuse” to form larger nuclei (helium, etc.), releasing energy

• Basic reaction:

4H He + 2e+ + 2γ + 2νwhere

e+ is a positron (anti-particle of the electron)

γ is a gamma-ray photon

ν is a “neutrino”

•Most of the energy released is carried by the positrons and gamma rays

4 1H (protons)

4He

Nuclear Fusion

Hydrogen fuses to Helium

Start: 4 + 2 protons End: Helium nucleus + neutrinos Hydrogen fuses to Helium

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Iron (Fe) weighs lessper proton than anything else

Each proton in Uranium weighs more

Each proton in hydrogenweighs more

Why should the fuse?Why does a ball roll downhill?To minimize energy!