RELATIVISTIC EFFECTS

According to the system of classical mechanics that was developed by Newton and the others, mass is an immutable property of matter; it can be changed in size, shape, or state but it can neither be created nor be destroyed. Although this law of conservation of mass seems to be true for the world that we can perceive with our senses, it is in fact only a special case for conditions of large masses and slow speeds.

In the submicroscopic world of the atom, where masses are measured on the order of kg ,where distances are measured on the order of m ,and where velocities are measured in terms of the velocity of light, classical mechanics is not applicable. Einstein in his special theory of relativity, postulated that the velocity of light in a vacuum is constant at 3 × m/s relative to every observer in any reference frame. He also postulated that the speed of light is an upper limit of speed that a material body can asymptotically approach, but never can attain.

Furthermore, according to Einstein, the mass of a moving body is not constant, as was previously thought, but rather a function of the velocity with which the body is moving. As the velocity increases, the mass increases, and when the velocity of the body approaches the velocity of light, the mass increases very rapidly. The mass m of a moving object whose velocity is v is related to its rest mass m0 by the equation:

where c is the velocity of light, 3 × m/s Example 1 Compute the mass of an electron moving at 10% and 0.99% of the speed of light. The rest mass of an electron is 9.11 x kg. Solution At v = 0.1c

and at v = 0.99 c , Example1 shows that whereas an electron suffers a mass increase of only 0.5% when it is moving at 10% of the speed of light, its mass increases about sevenfold when the velocity is increased to 99% of the velocity of light. Kinetic energy of a moving body can be thought of as the income from work put into the body, or energy input, in order to bring the body up to its final velocity. Expressed mathematically, we have

However, the expression for kinetic energy in Eqs. (2. 3) and (2 However, the expression for kinetic energy in Eqs. (2.3) and (2.5) is a special case since the mass is assumed to remain constant during the time that the body is undergoing acceleration from its initial to its final velocity. If the final velocity is sufficiently high to produce observable relativistic effects (this is usually taken as v ≈ 0.1c = 3 × m/s, then Eqs. (2.3) and (2.5) are no longer valid. As the body gains velocity under the influence of an unbalanced force, its mass continuously increases until it attains the value given by Eq. (2.4). This particular value for the mass is thus applicable only to one point during the time that body was undergoing acceleration.

Equations (2.2) and (2.5) assume the force to be constant and therefore are not applicable to cases where relativistic effects must be considered. One way of overcoming this difficulty is to divide the total distance r into many smaller distances, , ,……, as shown in Figure 2-1, multiply each of these small distances by the average force exerted while traversing the small distance, and then sum the products. This process may be written as:

dW = F dr (2.7) and symbolized by If r represents distance, dr represents an infinitesimally small distance and the differential of work done, which is the product of the force and the infinitesimally small distance, is given by dW = F dr (2.7)

This sum is indicated by the mathematical notation Since acceleration is defined as the rate of change of velocity with respect to time: where v1 and v2 are the respective velocities at times t1 and t2. Then Eq. (2.1) maybe written as

Using the differential notation, we have: This is the expression of Newton’s second law of motion for the nonrelativistic case where the mass remains constant. Newton’s second law states that the rate of change of momentum of an accelerating body is proportional to the unbalanced force acting on the body. For the general case, where mass is not constant, Newton’s second law is written as”

Substitution of the value of f from Eq. (2.12) into Eq. (2.8) gives: Since v = dr/dt, Eq. (2.13) can be written as: And substituting , we have

Differentiating the term in the parenthesis gives: Now, multiply the numerator and denominator of the first term in Eq. (2.16) by to obtain

The integrand in Eq. (2.18) is almost in the form

Where and To convert Eq. (2.18) into the form for integration given by Eq. (2.19), it is necessary only to complete du. This is done by multiplying the integrand by and the entire expression by in order to keep the total value of Eq. (2.18) unchanged. The solution of Eq. (2.18), which gives the kinetic energy of a body that was accelerated from zero velocity to a velocity v, is

Where Equation (2.20) is the exact expression for kinetic energy and must be used whenever the moving body experiences observable relativistic effects.

Example 2:What is the kinetic energy of the electron in Example1 that travels at 99% of the velocity of light? Solution

The relativistic expression for kinetic energy given by Eq. (2 The relativistic expression for kinetic energy given by Eq. (2.20) is rigorously true for particles moving at all velocities while the nonrelativistic expression for kinetic energy, Eq. (2.3), is applicable only to cases where the velocity of the moving particle is much less than the velocity of light. It can be shown that the relativistic expression reduces to the nonrelativistic expression for low velocities by expanding the expression in Eq. (2.20) according to the binomial theorem and then dropping higher terms that become insignificant when v « c . According to the binomial theorem

The expansion of According to Eq. (2.21), is accomplished by letting a = 1, b = −β2, and n = −1/2. Since β = v/c , then, if v « c, terms from β and higher will be insignificantly small and may therefore be dropped. Then, after substituting the first two terms from Eq. (2.22) into Eq. (2.20), we have: 4 which is the nonrelativistic case. Equation (2.3) is applicable when v « c

In nonrelativistic cases, the increase in velocity is directly proportional to the square root of the work done on the moving body or, in other words, to the kinetic energy of the body. In the relativistic case, the velocity increase due to additional energy is smaller than in the nonrelativistic case because the additional energy serves to increase the mass of the moving body rather than its velocity. This equivalence of mass and energy is one of the most important consequences of Einstein’s special theory of relativity. According to Einstein, the relationship between mass and energy is:

where E is the total energy of a piece of matter whose mass is m and c is the velocity of light in vacuum. The principle of relativity tells us that all matter contains potential energy by virtue of its mass. It is this energy source that is tapped to obtain nuclear energy. The main virtue of this energy source is the vast amount of energy that can be derived from conversion into its energy equivalent of small amounts of nuclear fuel.