Question

state coloumb's law. Deduce coulomb's law from gauss's theorem and express it in vector form

Answer 1

Coulomb's law states that: The magnitude of the electrostatic force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of charges and inversely proportional to the square of the distance between them.Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the curl of E (u can refer Helmholtz decomposition and Faraday's law). However, Coulomb's law can be proven from Gauss's law if it is assumed, in addition, that the electric field from a point charge is spherically-symmetric (this assumption, like Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion). Taking S in the integral form of Gauss's law to be a spherical surface of radius r, centered at the point charge Q, we have
\oint_{S}\mathbf{E}\cdot d\mathbf{A} = Q/\varepsilon_0
By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is
4\pi r^2\hat{\mathbf{r}}\cdot\mathbf{E}(\mathbf{r}) = Q/\varepsilon_0
where \hat{\mathbf{r}} is a unit vector pointing radially away from the charge. Again by spherical symmetry, E points in the radial direction, and so we get
\mathbf{E}(\mathbf{r}) = \frac{Q}{4\pi \varepsilon_0}\frac{\hat{\mathbf{r}}}{r^2}
which is essentially equivalent to Coulomb's law. Thus the inverse-square law dependence of the electric field in Coulomb's law follows from Gauss's law.

Answer 2

Coulomb's Law is one of the basic ideas of electricity in physics. The law looks at the forces created between two charged objects. As distance increases, the forces and electric fields decrease. The force between the objects can be positive or negative depending on whether the objects are attracted to each other or repelled.  

When you have two charged particles, an electric force is created. If you have larger charges, the forces will be larger. It's a formula that measures the electrical forces between two objects.

F=kq1q2/r2

"F" is the resulting force between the two charges. The distance between the two charges is "r" also known as radius of seperation  

The "q1" and "q2" are values for the amount of charge in each of the particles and K is constant