Dimensional Dilemma!

Consider an \(n\)-dimensional solid sphere of radius \(R\), with uniform charge-density and a total charge \(Q\). A charged particle \(q\) is kept at a distance of \(r\) from the center. For \(r<R\), what is the Net Coulombic Force experienced by the particle?

If it is in the form of

\[\displaystyle\vec{F_{c}}= \left(\dfrac{\color{blue}{m}}{\color{blue}{p}.\varepsilon_{0}}\right).\left(\dfrac{(\color{blue}{a}+2)!}{\left(\Gamma\left(\dfrac{\color{blue}{b}}{\color{blue}{c}}\right)\right)^{\color{blue}{x}}.\color{blue}{w^{y}}}\right)\]

Find \(\color{blue}{ m+p+a+b+c+w+x+y}\).

Details and Assumptions:

  • \(n=12\), \(Q=10 \text{ C}\), \(q=1.4 \text{ C}\), \(r=5 \text{ cm}\), \(R=13\text{ cm}\).

  • All the letters represent integers and are not necessarily distinct. \(m\) and \(n\), \(b\) and \(c\) are coprime.

  • \(\varepsilon_{0}\) is the constant of permittivity of free space.


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