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Technical Health Physics Papers - The Mathematics of Radiation Protection

E=MC2

Article written by Chris Robbins of Grallator Ltd

You can also download a PDF of E=MC2

This equation is probably the most famous in the whole of physics. It relates the energy content of matter to its mass through the speed of light squared. As this is such a large number it means that 1kg of matter contains the equivalent energy of about 90 thousand million million joules - enough to boil a cube of water with a length, width and height each of about 600 metres. If we consider the energy output from the sun (3.846×1026 W) and run the calculation backwards we deduce that the sun converts 4.3 million tonnes of matter into energy each second! As the sun has a mass of about 2×1021 million tonnes we’re in no immediate danger of it losing all its mass this way.

The formal derivation of the equation comes from the 4-dimensional mathematics of special relativity, but once "out of the bag", other, less formal derivations were constructed. One of the most transparent is shown below.

Consider a carriage with a catapult loaded with an object as shown below

emc2_400

When the catapult releases the ball it will fly to the right, while the carriage will move to the left. This is the rocket principle and it arises from conservation of momentum or, if you like, Newton's Third Law (for every action there is an equal and opposite reaction). When the ball hits the upright at the end of the carriage both the ball and the carriage will come to a halt. No net velocity will be imparted (in an ideal scenario) as this would allow locomotion without reaction -you could fire lots of balls and quickly collect them up and re-use them thus moving the carriage without any apparent external force. This is like lifting yourself off the ground by pulling at your shoe laces -patently impossible! Also, if you consider the centre of gravity of the carriage/ball system you will find that it hasn't moved as again this would mean a net force had been applied.

Imagine now that the catapult is replaced by a low intensity laser which can be turned down to release just a single photon. It is known that photons carry energy and momentum, with the relationship being

emc_1_400   
E being the energy, p its momentum and c is the speed of light. To conserve momentum the carriage must move in the opposite direction at some speed v. If he mass of the carriage is M then conservation of momentum means that

emc_2_400
which I will re-write in terms of E as 

emc_3_400

Now when the photon reaches the far end of the carriage it is reabsorbed and the carriage comes to rest. As the carriage has moved slightly to the left and the centre of gravity of the system hasn't moved, then we can conclude that the photon has transferred an equivalent mass m from one end of the carriage to the other. If the photon travels a distance L and the carriage a distance x, then this means that by considering the moments about the centre of mass

emc_4_400

Re-arranging (4) gives an expression for M as

emc_5_400

As the distance travelled by the photon is L and its speed is c then the time taken to travel is

emc_6_400

The carriage speed is v and as velocity equals distance over time I can write

emc_7_400

where the value of t given in (5) has been substituted.

I can now use (5) for M and (7) for v in (3) to give

emc_8_400

where m is the effective mass transferred.

Chris Robbins
28 August, 2008



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