# How to compute electrical charge over semi circumference with linear density $\rho_l = k\sin\phi$ without using calculus

The semi circumference has a radio R is electrified with linear density $\rho_l = k \sin \phi$ ($k$ = constant). Determine its total charge. Answer: $2kR$

My answer:

Given: $dQ = \rho_l\cdot R \cdot d\phi$

$Q = \int_{0}^{\pi}R\cdot k\cdot \sin\phi \,d\phi\Rightarrow Q = R.k\cdot\underbrace{(-cos(\phi)|_{0}^{\pi})}_{=2} = 2kR$

Very easy honestly.

But my questions is how can I solve this without calculus tools?

## 1 Answer

I don't think you can solve such problems without the aid of calculus. At least, whenever uniform densities are involved in such a manner, I've never heard of methods that do not involve some form of calculus. Also, all the fundamental laws of classical electrodynamics are either in integral or differential forms. One can try an approach of looking at uniform densities such as these as a set of discrete charges with equal amount of spacing between them and the space between the charges tends to zero. This again would involve using limits and thus calculus. So I don't think there's any escape from calculus in such problems, at least, not that I'm aware of.