Sunday 19 November 2017

electrostatics - Coulomb's law with an $r^3$, not $r^2$, in the denominator




I am reading an older physics book that my professor gave me. It is going over Coulomb's law and Gauss' theorem. However, the book gives both equations with an $r^3$, not $r^2$, in the denominator. Can somebody please explain why it is given as r^3? An image is attached for reference.


Also for equation 1-24, can somebody please explain how the middle side is equal to the right side with the del operator?


enter image description here



Answer



It does give "Coulomb's law" with $\frac{1}{r^3}$, it gives it in its proper vectorial form $$ \vec E \propto \frac{\vec r}{r^3}$$ which, when taking the absolute values, yields the form you are probably more familiar with $$ E \propto \frac{1}{r^2}$$ since $\lvert \vec r \rvert = r$.


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