Question

Basic Coulomb's law tells us that the electric field of a point charge q, E=k_e q/r², scales with distance r as 1/r²: when the distance increases by a factor of 2, the magnitude of the field decreases by a factor of 4. We can write this as E(r) ∝ 1/r². An interesting and practically important question is how the electric field of a group of charges scales with distance (asymptotically) far away from the group. The figure below depicts 3 different groups of charges; for each of these 3 cases we will be interested in the magnitude of the electric field at point P, distance r away from the group center. The exact analysis shows that in all cases, in the limit of r ≫ a, the distance scaling takes place as E(r) ∝ 1/rⁿ, where power n is an integer. You are asked to deduce the values of power n for all 3 configurations from the numerical calculations as described below.

P P Р (а) (b) (с) q |ь Б- а а а а а

Given specific value of parameter r =14 a, calculate the ratio E(r)/E(2r) of magnitudes of the electric fields at point P for two different distances, r and 2r, in each of the configurations. This numerical ratio should be approximately equal to 2ⁿ for scaling E(r) ∝ 1/rⁿ, and that is how you determine these integer values of n.

Configuration (a), where the total charge of the group is not zero.

E(r)/E(2r)= ________ ,n= _________ .

Configuration (b), where the total charge of the group is zero. This is a so-called electric dipole.

E(r)/E(2r)= ________ ,n= _________ .

Configuration (c), where the total charge of the group is zero. This is a so-called electric quadrupole.

E(r)/E(2r)= ________ , n= _________ .

Answer 1

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