Pair production question - photon to electron/positron

[Brianrofl]

Homework Statement

http://puu.sh/brbpb/3c7573fa32.png

Homework Equations

E = (mc^2 +K + mc^2 + K)
P = E/c
E = mc^2

The Attempt at a Solution

The book says that the momentum/kinetic energy of the electron and positron produced in a pair production is so small that it can be assumed that the electron only holds rest energy.

I just don't know. The question states that they have momentum, but how am I supposed to find that when I'm given so little information? All I know is that the photon has at least 1.02MeV, but how am I supposed to know how much extra energy went towards increasing the electron's momentum?

Thanks.

 

[vela]

Why don't you start by doing what the hint says? Write down the equations for conservation of energy and momentum.

 

[Brianrofl]

Why don't you start by doing what the hint says? Write down the equations for conservation of energy and momentum.

I've gone through so much scratch paper and just can't get it.


Momentum equation:

Initial electron is at rest, has no momentum

Three electrons have same momentum afterwards.

P = 3mv
P = E/c, so
E = 3cmv

Energy conservation equation - third electron has kinetic energy now, but rest energy not included since it was not created by the photon.

E = (mec^2 + K) + (mec^2 + K) + K

3cmv = 2mec^2+3k

And from there I just get a ton of ridiculous values for v

 

[vela]

I've gone through so much scratch paper and just can't get it.Momentum equation:

Initial electron is at rest, has no momentum

Three electrons have same momentum afterwards.

P = 3mv

This isn't correct. At relativistic speeds, $p\ne mv$.

P = E/c, so

This is only true for massless particles.

E = 3cmv

Energy conservation equation - third electron has kinetic energy now, but rest energy not included since it was not created by the photon.

E = (mec^2 + K) + (mec^2 + K) + K

This is okay, but I would say
$$ E_\gamma + m_ec^2 = 3E_e$$ where $E_\gamma$ is the energy of the photon and $E_e$ is the energy of an electron or positron in the final state.

With relativity problems, it's easier to work with energy and momentum. Once you have solved for those, you can then find velocities if needed. (Note that you don't need to find the velocity in this problem.)

It's also useful to use the relationship $E^2 - (pc)^2 = (mc^2)^2$. For an electron or positron, you'd have $E_e^2 - (p_ec)^2 = (m_ec^2)^2$. (So you can see that E/c isn't equal to p for the electron.) A photon's invariant mass is 0, so the relationship reduces to $E_\gamma^2 = (p_\gamma c)^2$ or $E_\gamma = \lvert p \rvert c$.

3cmv = 2mec^2+3k

And from there I just get a ton of ridiculous values for v