The following problems consider the scalar form of Coulomb\'s law, which describ
ID: 1770705 • Letter: T
Question
The following problems consider the scalar form of Coulomb's law, which describes the electrostatic force between two point charges, such as electrons. It is given by the equation F(r)=keq1q2r2,F(r)=ke|q1q2|r2, where keke is Coulomb's constant, qiqi are the magnitudes of the charges of the two particles, and rr is the distance between the two particles
To simplify the calculation of a model with many interacting particles, after some threshold value r=R,r=R, we approximate ff as zero.
Explain the physical reasoning behind this assumption.
What is the force equation?
Evaluate the force FF using both Coulomb's law and our approximation, assuming two protons with a charge magnitude of 1.6022×1019coulombs (C),1.6022×1019coulombs (C), and the Coulomb constant ke=8.988×109Nm2/C2ke=8.988×109Nm2/C2 are 11 m apart. Also, assume R<1m.R<1m. How much inaccuracy does our approximation generate? Is our approximation reasonable?
Is there any finite value of RR for which this system remains continuous at RR?
Explanation / Answer
(1) Coulomb's force equation is based on the force of attraction between the two static charges, which is called the Coulomb's law of force in electrostatics which states that the electrostatic force between any two point charges q1 and q2 separated by a distance r12 is proportional to the product of the charges and inversely proportional to the square of the distance between them,
F = k x -q1q2/r122 , where k = 9 x 109 and the negative sign means attraction between the two charges.
(2) By the coulomb's law of force and the given approximation, to calculate the force FF between two protons with given dimensions, since the like charges repel each other, but there exist an infinitesimal attraction which causes the repulsion between them, then taking it into consideration we can say that the force FF is equal to
FF = k2 |q|2/r4, this is because since one F = ke|q1q2|/r2 and since we have to calculate the force FF between two protons q1 will be change to |q1|2 and r2 will remain as r2.
Therefore, the force FF = (9 x 109) x |1.6022 x 10-19|2 / (11)2
FF = 23.10340356 x 10-38+9/ 121
FF = 0.1909 x 10-29 = 1.909 x 10-30 C2/m4. Which is the required value of force FF, but after the threshold frequency r = R and r = R, the values of FF are zero, which means that till the threshold distance or till the minimum distance between the two particles, the minimum distance with which they are separated the force has a finite value, but if the distance between the particles is greater than the threshold distance, then the force becomes infinite. Assuming that R and R is less than 1m, we can say that the approximation is accurate to the actual value of the force using the coulomb's law because since we can assume the threshold smaller than 1m or 1cm we can actually somehow get the same value as that of computed value of coulomb's law.
The approximation is truly reasonable since it uses the concept of threshold distance which is itself an insight to what exactly constitutes a force between two charges, the condition of infinity can be taken into consideration, since the zero distance signfies the infinite force between the two charges.
For the system to be continuous, the value of RR must less than 1m, because this distance is approximated value of the distance between two particles, that is this measure of distance is the maximum separation between two charges and beyond this the force will be infinite.
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