A Discussion On The Square Inverse Ratio Law在平方反比定律的探讨

A Discussion On The Square Inverse Ratio Law在平方反比定律的探讨 A Discussion On The Square Inverse Ratio Law Jianbo Deng Physics Department of Lanzhou University Lanzhou 730000, P. R. China Abstract According to basic laws in classical mechanics, this article discusses the active form of the basic forces in static state case. It shows that the abstract concept such as a point charge (itself) can lead to the result of a square inverse ratio law. As soon as we have the concept of a point charge, we should have the active form of force of the square inverse ratio written down. A further discussion shows that the square inverse ratio form of force is somehow a basic nature of long-range force. I. Introduction Gravity and Electromagnetic interaction are two basic forces in classic Physics. It is very interest that these two forces all behave the form of square inverse ratio in static case. This article gives an explanation based on the basic properties of force in classical mechanics, and the supposition that the basic elements on which interaction takes place can be abstracted to a point charge. II. The square inverse ratio form Suppose there exists a certain point quantity kon which interaction takes place. So we can use k and k to describe the active form of force. Define r as a position 12 vector from k to k. The active form of force which k produces on k is: 2121 F= F(k, k, r) , (1) 212112 where k is a kind of quantity with a certain substance, so it is reasonable to define that k is plus able. According to Newton’s Third law, we know k and k should be at the same 12 position, and be symmetrically put together. However, r is different from k. So formation (1) should be possibly like: ˆ ， (2) F,f(k,k,r)n2112 k,k,k,kwhere we have used the property that demanded by the synthesize 1221 ,rule of force, the mathematical symbol could represent the mathematical symbols ˆnfor addition， and multiplication; and is a unit victor. ， Again considering Newton’s Third law, F should be symmetrically in the direction of r. Thus (2) becomes ˆF,f(k,k,r)r (3) 211221 We’ll try to analysis (3) by the Taylor series, which is found out to be a powerful method. k,k,0First, let us open up by the Taylor series at ; let f(k,k,r)1212 ,,k,k, then at a certain r, (3) becomes 12 2,f1,f2ˆ,F,[f(,r)，(k,k)，(k,k)，,,,]r . (4) 21121221,,0,,0,,02,,2,, k,k,0F,0We know that at , , then we get 1221 f(,r),0, , (5) ,,0 so (4) becomes: 2,f1,f2ˆF,[(k,k)，(k,k)，,,,]r . (6) 21121221,,0,,02,,2,, ,Here in (6) if is +, then it cannot satisfy both the property of force and the plus character of k at the same time (For example, F should be doubled if one of k 1 f, ,and kgets doubled). When is , only term in (6) is reasonable for both ，2 ,,0,, the property of force and the plus character of k at the same time. Let ,f(,r),,(r),, Thus: ,,0,, ˆF,,(r)kkr . (7) 211221 ,(r)The next discussion is to find out the detailed expression of . It is 1u,,(r),0h(u),,(r)r,,reasonable considering , . Then orders , , we r h(0),0r,,know that at , . So (7) becomes: ˆF,kkh(u)r . (8) 211221 h(u)u,0Now we can go on open up by the Taylor series at , then: 12ˆF,kk[h'(0)u，h''(0)u，,,,]r 2112212 111ˆkkhh,['(0)，''(0)，,,,]r . (9) 12212r2r One of the main purposes in this article is to show that, the abstract concept such as a point charge (itself) can lead to the result of a square inverse ratio law. One can try to discuss all the items in (9) one by one, and will find out that they can not lead to the abstract of a point charge concept, except for the square inverse ratio item. Here we will show the exact discussion of this procedure for the square inverse ratio item in (9). 1C,h''(0), then Order 2 kk12ˆF,Cr . (10) 21212r Suppose K and K are two groups of k whose distribution is spherical in 12 symmetry, we may write this expression: ~~~~2K,rdV,rrdr,(),()4,,, , (11) 11111111 VV11 ~~~~2K,rdV,rrdr,(),()4,,, , (12) 22222222 VV22 ~~,(r),(r)where and are the spherical distributed densities. 1122 RIf is a vector from the center of sphere 2 to the center of sphere 1, then 21 from (10) we know the active force that K produce on K is: 21 ~~~~,,CrrrRr()()()，,121212FdVdV,, . (13) 21123~~VVrRr，,121212 ,Usually one will expect the result of the above integral related with and r. This expectation would be right if the above integral were not for the square inverse V:V,,ratio item; but this time, a long careful calculation of (13) in the case of 12only simply leads to: KKKK1212ˆF,CR,CR , (14) 21212132RR2121 which is very surprising! From these results, we learn that the square inverse ratio form is what we want. When k is a spherical in symmetric distribution, then active formation of two groups of k is the same as that of two point charges. It will result in infinite inter-energy to regard k as a point charge. The most basic distribution of k in nature is spherical in symmetry. If the active form were not a square inverse ratio, we wouldn’t be able to abstract k to be a point charge. This is contrary to my leader supposition. Therefore the k active form of force between two point charges should be chosen as a square inverse ratio. Now we can say (1) is finally settled as: kk12ˆF,Cr , (15) 21212r21 kk12ˆF,rCi.e. . (16) 2r III. Further discussions and comments As the applications of our above result, the two basic forces (which is most related to our daily life) in classic Physics are discussed as examples bellow. (a). Gravity Thanks to the square inverse ratio, things become much more simple since we can abstract any a spherical celestial body into a point to discuss the static force. A well know example is the experiment close to the Earth’s surface, it gets: kke1ˆF,CR,mg , 21e12Re where R is the radius of the Earth. Since the downward acceleration g is a constant, e m,kwe know that the mass . Then by choosing a suitable unit, (16) becomes: mm12ˆFr,,G, 2r this is well known as Newtonian Gravitation Law. (b). Electrostatic action A charged particle can be also treated in the same way above. Define k=q , then (16) can be written as: qqqq11212ˆˆF,r,rC, 21224,,rr0 we get Coulomb’s Law. Here again we would like to express our gratitude to the power of the square inverse ratio, for because of which we may live in a world of peace and harmony. Since the negative charged electrons in an atom could be almost treated as spherically distributed, then to the out side world, they can be ‘shrunk’ to the center to ‘balance’ the effect of the positive charged nucleus. It is also important to know that, the understanding of the conservation law of charge in electromagnetism is based on the above discussion, because we detect ‘charge’ by force. At the end of this article, we would like to put the form of the square inverse ratio in this way, that is we are lucky to have this form, because of which we can abstract a point concept to discuss the long-range forces, to live in peace and harmony, to find out the conservation law of charge etc, even before we realized it. The square inverse ratio form of force leads us to the thought of these words: simple, beautiful and harmonic. Now we would like to say that the square inverse ratio form of force is somehow a basic nature of long-range force. The Laws are carefully made for us.