Obtain Coulomb’s law from Gauss’s law.
Obtain Coulomb’s law from Gauss’s law.
As shown in figure, consider a point charge $+q$ kept at $\mathrm{O}$.
A Gaussian surface ' $\mathrm{S}$ ' is shown in figure includes charge $q$.
Consider surface area $\overrightarrow{d s}$ at point $\mathrm{P}$. Here, $\overrightarrow{\mathrm{E}} \| \overrightarrow{d s}$ and so that $\theta=0^{\circ}$.
According to Gauss's law,
$\phi=\frac{q}{\varepsilon_{0}}$ $\therefore \int \overrightarrow{\mathrm{E}} \cdot \overrightarrow{d s}=\frac{q}{\varepsilon_{0}}$ $\therefore \int \mathrm{E} \cdot d s \cos 0^{\circ}=\frac{q}{\varepsilon_{0}}[\because \overrightarrow{\mathrm{E}} \| \overrightarrow{d s}]$ $\therefore \mathrm{E} \int d s=\frac{q}{\varepsilon_{0}}\left[\therefore \cos 0^{\circ}=1\right]$ $\therefore \mathrm{E} \times 4 \pi r^{2}=\frac{q}{\varepsilon_{0}}\left[\therefore \int d s=4 \pi r^{2}\right]$ $\therefore \mathrm{E}=\frac{q}{4 \pi \varepsilon_{0} r^{2}}$ $\therefore \frac{\mathrm{F}}{q}=\frac{q}{4 \pi \varepsilon_{0} r^{2}}$ $\therefore \mathrm{F}=\frac{\mathrm{Kq} q_{0}}{r^{2}}$ $\mathrm{This}$ is Coulomb's law.
Similar Questions
Consider a sphere of radius $R$ with charge density distributed as :
$\rho(r) =k r$, $r \leq R $
$=0$ for $r> R$.
$(a)$ Find the electric field at all points $r$.
$(b)$ Suppose the total charge on the sphere is $2e$ where e is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution.
Consider a sphere of radius $R$ with charge density distributed as :
$\rho(r) =k r$, $r \leq R $
$=0$ for $r> R$.
$(a)$ Find the electric field at all points $r$.
$(b)$ Suppose the total charge on the sphere is $2e$ where e is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution.
Obtain the expression of electric field by charged spherical shell on a point outside it.
Obtain the expression of electric field by charged spherical shell on a point outside it.
A long, straight wire is surrounded by a hollow, thin, long metal cylinder whose axis coincides with that of wire. The wire has a charge per unit length of $\lambda$, and the cylinder has a net charge per unit length of $2\lambda$. Radius of the cylinder is $R$
A long, straight wire is surrounded by a hollow, thin, long metal cylinder whose axis coincides with that of wire. The wire has a charge per unit length of $\lambda$, and the cylinder has a net charge per unit length of $2\lambda$. Radius of the cylinder is $R$
A non-conducting solid sphere of radius $R$ is uniformly charged. The magnitude of the electric field due to the sphere at a distance $r$ from its centre
A non-conducting solid sphere of radius $R$ is uniformly charged. The magnitude of the electric field due to the sphere at a distance $r$ from its centre
- [IIT 1998]
The electric intensity due to an infinite cylinder of radius $R$ and having charge $q$ per unit length at a distance $r(r > R)$ from its axis is
The electric intensity due to an infinite cylinder of radius $R$ and having charge $q$ per unit length at a distance $r(r > R)$ from its axis is