累進析句法整段範例:普通物理電學篇

(Source: Calculus-Based Physics II, 2006., by Allen B. Downey, GNU Free Documentation License 授權。)

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Charge is a property of matter. There are two kinds of charge, positive "+" and negative "-". An object can have positive charge, negative charge, or no charge at all. A particle which has charge causes a force-per-charge-of-would-be-victim vector to exist at each point in the region of space around itself. The infinite set of force-per- charge-of-would-be-victim vectors is called a vector field. Any charged particle that finds itself in the region of space where the force-percharge-of-would-be-victim vector field exists will have a force exerted upon it by the force-per-charge-of-would-be-victim field. The force-per-charge-of-would-be-victim field is called the electric field. The charged particle causing the electric field to exist is called the source charge. Regarding jargon: A charged particle is a particle that has charge. A charged particle is often referred to simply as "a charge."

The source charge causes an electric field which exerts a force on the victim charge. The net effect is that the source charge causes a force to be exerted on the victim. While we have much to discuss about the electric field, for now, we focus on the net effect, which we state simply (neglecting the "middle man", the electric field) as, "A charged particle exerts a force on another charged particle." This statement is Coulomb’s Law in its conceptual form. The force is called the Coulomb force, a.k.a. the electrostatic force.

Note that either charge can be viewed as the source charge and either can be viewed as the victim charge. Identifying one charge as the victim charge is equivalent to establishing a point of view, similar to identifying an object whose motion or equilibrium is under study for purposes of applying Newton's 2nd Law of motion, [formula].* In Coulomb's Law, the force exerted on one charged particle by another is directed along the line connecting the two particles, and, away from the other particle if both particles have the same kind of charge (both positive, or, both negative) but, toward the other particle if the kind of charge differs (one positive and the other negative). This fact is probably familiar to you as, "like charges repel and unlike attract.

The SI unit of charge is the coulomb, abbreviated C. One coulomb of charge is a lot of charge, so much that two particles, each having a charge of +1 C and separated by a distance of 1 meter, exert a force of 9x109N, that is, 9 billion newtons on each other.

This brings us to the equation form of Coulomb's Law which can be written to give the magnitude of the force exerted by one charged particle on another as [formula1(1-1)], where: k = [formula2], a universal constant called the Coulomb constant, q1 is the charge of particle 1, q2 is the charge of particle 2, and r is the distance between the two particles.


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The user of the equation (we are still talking about equation 1-1, [formula]) is expected to establish the direction of the force by means of "common sense" (the user's understanding of what it means for like charges to repel and unlike charges to attract each other).

While Coulomb's Law in equation form is designed to be exact for point particles, it is also exact for spherically symmetric charge distributions (such as uniform balls of charge) as long as one uses the center-to-center distance for r.

Coulomb's Law is also a good approximation in the case of objects on which the charge is not spherically symmetric as long as the objects' dimensions are small compared to the separation of the objects (the truer this is, the better the approximation). Again, one uses the separation of the centers of the charge distributions in the Coulomb's Law equation.

Coulomb's Law can be written in vector form as [formula], where F12 is the force "of 1 on 2", that is, the force exerted by particle 1 on particle 2, r12 is a unit vector in the direction "from 1 to 2", and k, q1, and q2 are defined as before (the Coulomb constant, the charge on particle 1, and the charge on particle 2 respectively).

Note the absence of the absolute value signs around q1 and q2. A particle which has a certain amount, say, 5 coulombs of the negative kind of charge is said to have a charge of -5 coulombs and one with 5 coulombs of the positive kind of charge is said to have a charge of +5 coulombs and indeed the plus and minus signs designating the kind of charge have the usual arithmetic meaning when the charges enter into equations. For instance, if you create a composite object by combining an object that has a charge of q1=+3C with an object that has a charge of q2=-5C, then the composite object has a charge of q=q1+q2, q=+3C+(-5C), q=-2C.


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Note that the arithmetic interpretation of the kind of charge in the vector form of Coulomb's Law causes that equation to give the correct direction of the force for any combination of kinds of charge. For instance, if one of the particles has positive charge and the other negative, then the value of the product q1q2 in equation 1-2[formula] has a negative sign which we can associate with the unit vector. Now -r12 is in the direction opposite "from 1 to 2" meaning it is in the direction "from 2 to 1." This means that F12, the force of 1 on 2, is directed toward particle 1. This is consistent with our understanding that opposites attract. Similarly, if q1 and q2 are both positive, or both negative in [formula] then the value of the product q1q2 is positive meaning that the direction of the force of 1 on 2 is r12 (from 1 to 2), that is, away from 1, consistent with the fact that like charges repel.

We've been talking about the force of 1 on 2. Particle 2 exerts a force on particle 1 as well. It is given by [formula]. The unit vector r21, pointing from 2 to 1, is just the negative of the unit vector pointing from 1 to 2: r21= -r12. If we make this substitution into our expression for the force exerted by particle 2 on particle 1, we obtain: [formula1] [formula2].

Comparing the right side with our expression for the force of 1 on 2 (namely, [formula]), we see that F21=-F12.

So, according to Coulomb's Law, if particle 1 is exerting a force F12 on particle 2, then particle 2 is, at the same time, exerting an equal but opposite force F21 back on particle 2, which, as we know, by Newton's 3rd Law, it must.