What kinds of forces can we have in relativity? Instantaneous action at a distance?

What kinds of forces can we have in relativity?•Instantaneous action at a distance?

2

ˆkqQrF

r

2

ˆGmMrF

r

These are no good because instantaneous effects violate relativity•No problem, we already know about electric and magnetic fields

Q q

F q qu

Gravity is very hard:Einstein’s General Theory of

Relativity

Magnetic forceElectric force

Newton’s Laws and ForceNewton’s Second Law in non-relativistic physics:

F ma dv

mdt

dmv

dt

1 2F F

1 20 F F

1 2dp dp

dt dt

dp

dt

F

Newton’s Third Law:

1 2 ,d

p pdt

1 2 constantp p

Conservation of momentum comes from Newton’s Third Law and One Version of Newton’s Second Law

1 20 F F dp

Fdt

F ma

Relativity

Cyclotron Motion

q

Consider a charged particle moving in a magnetic field:•Magnetic field points out of the slide

dp

dt

F qu

•Force keeps speed the same is constant

u

F

dm u

dt du

mdt

•Centrifugal acceleration:2du u

dt R

R 2quB m u RqBR m u

p qBR

B in TeslaR in metersq in Coulombsp in kgm/s

Worku pc

c E

22 2 2 2E c p mc

dpF

dt

•Take time derivative of the first one•Solve for dE/dt•Substitute the second•Substitute the third•Rewrite the velocity•Integrate over time

22 2 0dE dp

E c pdt dt

2dE c p dp

dt E dt

dp

udt

u F

drF

dt

dE F dr

E W

W E F d

Sample ProblemAn electron (m = 511.0 keV/c2) at rest is placed in an electric field of

magnitude 100.0 V/cm. How long does it take, and how far does it go, before it reaches a velocity of v = 0.500c?

F dp dt

E F d

•Work formula:

F q 410 eV / m 10 keV / m

Fd E 2f i mc

2

11, 1.155

1 0.500i f

20.155mc

0.155 511.0 keV

10 keV/md 7.94 m

•Momentum formula: Ft p 0fp f mu 0.577mc20.577mc

tFc

8

0.577 511 keV

10 keV/m 3.00 10 m/s

98.4 ns

Composite Objects and Invariant MassSuppose I have a box containing two objects of mass m moving with equal and opposite velocities u. What does momentum and energy look like for this whole object?

mu m u

tot 1 2E E E 2 2mc mc 22 mc

tot 1 2p p p ˆ ˆmu i i 0

•Looks exactly like a particle at rest of mass M = 2m.•If you Lorentz boost this object, the Energy/momentum will transform exactly the same way as a single object.•If we can’t see inside this object, we can’t tell it’s not a single object with this mass.

2M m 2M mMass of composite object is not the sum of its parts

Finding the Invariant MassSuppose I have a box containing many objects of various masses, moving at various velocities. What mass object M can have the same momentum and energy as the whole mess?

m1u1

m2u2 m3

u3

m4

u4

2tot

tot

i ii

i i ii

E m c

p m u

P

E2 2 2 2 4E c P M c

22 2 2 2tot totMc E c p

Effective Mass

Internal Energy and HeatSuppose I have a (solid) containing many atoms. Now I heat it up. Does the mass change?

Na

•The atoms start to move around•This increases the energy E of each atom•But the total momentum is still 0•The total energy E of the whole object increases•The invariant mass of the whole object increases

Cl Na Cl Na Cl

Cl Na Cl Na Cl Na

Na Cl Na Cl Na Cl

Cl Na Cl Na Cl Na

Heat

1 kg of water (specific heat 4.184 J/g/C) is heated from 0C to 100C. How much does the mass increase?

310 g 100 C 4.184 J/g/ CE 54.184 10 J

2E mc 2

Em

c

5

28

4.184 10 J

3.00 10 m/s

4.655 ng

Field Energy and MassSuppose I have a charged particle surrounded by electric fields. Does the field contribute to the mass?

•Electric fields have energy•Electric fields contribute to total energy•Electric fields contribute to total mass•The mass listed for a given particle includes this mass

q

Consider a hydrogen atom

p e•Proton and electron have cancelling charges•Partly eliminates the electric field•Decreases total energy•Decreases invariant mass Mass(H) < Mass(p) + Mass(e)

Binding energy counts like negative mass

Potential energyAny change in the potential energy of an object changes its mass•Heat•Electric energy•Chemical energy•Nuclear energy•[Gravitational energy is hard]

20E mc

There is nothing particularly special about nuclear energy•Other types of energy are too little to significantly affect the mass•For nuclear energy, calculating the mass difference may be the easiest way to find the total energy produced.

Massless:2

0E mc

Summary: Formulas you need

2x x

y y

z z

x

p p vE c

p p

p p

E E vp

2E mcp mu 22 2 2 2E c p mc

u pc

c E

E c p

u c

dpF

dt

W E F d

22 2 2 2tot totE c p Mc

p qBR

F q qu

End of material for Test 1

What’s the name of this Song?This day and age we're living in Gives cause for apprehension With speed and new invention And things like fourth dimension.

Yet we get a trifle weary With Mr. Einstein's theory. So we must get down to earth at times

Relax relieve the tensionAnd no matter what the progress Or what may yet be proved The simple facts of life are such They cannot be removed.

“As Time Goes By”Music and Lyrics by

Herman Hupfield

“When a man sits with a pretty girl for an hour, it seems like a minute. But let him sit on a hot stove for a minute and it's longer than any hour. That's relativity.”

Attributed to A. Einstein