Visual Physics

LIGHT and SPECIAL RELATIVITY

RELATIVISTIC ENERGIES

SUMMARY

Einstein’s 1^st postulate states that the laws of physics are the same for all observers in all inertial frames of reference. So, the laws of conservation of momentum and energy applied to isolated systems must be valid in all inertial frames of reference.

The mass m of a single object is taken as an intrinsic property of the object and is now considered an absolute quantity.

The velocity of a single object is measured w.r.t. to a fixed (stationary) frame of reference. The speed of the object is v.

Relativistic momentum

Relativistic energies: rest energy , total energy E, kinetic energy

Rest energy When an object is at rest , its energy is not zero, but instead given by Einstein’s famous equation (Note: and not E )

Total energy energy of a single object of mass m moving with speed v

Kinetic energy

Photon

photon is a massless particle

rest energy is zero

relativistic energy of a photon

linear momentum photon

Relativity factors gamma and beta

RELATIVISTIC ENERGIES

What is the energy of a single moving object?

The mass m of a single object is taken as an intrinsic property of the object and is now considered an absolute quantity. The velocity of the single object is measured w.r.t. to a fixed (stationary) frame of reference. The speed of the object is v.

The momentum p of a single object of mass m and speed v is

(1)

The total energy E of a single object of mass m and speed is

(2)

Equation 2 applies to any single object, for example, electron, proton, atom, molecule, basketball, star.

Equation 2 implies that even at rest the object possess energy by virtue of its mass m. This energy is called the rest energy

(3)

This rest energy can be converted into other forms such as kinetic energy or electromagnetic energy. When an object is not at rest, we can think of its total energy as the sum of its rest energy and its kinetic energy due to its motion

(4)

Thus, the kinetic energy K of the object is always a positive quantity and is given by equation 5

(5)

When but however much energy we give the object v < c.

At slow speed v << c, the kinetic energy is approximately equal to that given by the classical equation for the kinetic energy

(6) classical equation for kinetic energy

We can show that equation 6 is valid using the binomial approximation to express the factor

There are four parameters m, v, p and E that characterize the motion of an object. We can derive a useful expression for E in terms of p and m

(7)

(8)

Equation 7 is an important equation since linear momentum p is a more fundamental concept than kinetic energy. There is no conservation law of kinetic energy, whereas the law of conservation of linear momentum is inviolate as far as we know.

Rest energy of an electron

Rest energy of a proton

PHOTON

The photon is a type of elementary particle, the quantum of the electromagnetic field including electromagnetic radiation such as light, and the force carrier for the electromagnetic force. The mass of the photon is zero ; it always moves at the speed of light c within a vacuum.

total energy of a photon

Linear momentum photon

The energy of a photon is completely due to its motion, and not at all to its rest energy .

Any particle with zero mass must travel at the speed of light c in free space.

Example 1

Show that a particle of zero mass propagates at the speed of light.

Solution

Example 2

An electron is accelerated from rest by an electrical potential difference of 1.00 GV. What is the final velocity of the electron?

Solution

The accelerating voltage increases the kinetic energy of the electron

We cannot use the classical expression for kinetic energy.
Relativistic energies

The electron is accelerated to almost the speed of light. You will notice that relativistic calculations are more involved than their classical equivalent calculations.