Gauss’s Law and Its Applications

Which law is used to derive the expression for the electric field between two uniformly charged large parallel sheets with surface charge densities $\sigma$ and $-\sigma$ respectively:

Applying Gauss theorem, the expression for the electric field intensity at a point due to an infinitely long, thin, uniformly charged straight wire is

In a region of space, the electric filed $\overrightarrow{E}=E_{0}x\hat{\mathrm{i}}+E_{0}y\hat{\mathrm{j}}$, Consider imaginary cubical volume of edge $a$ with its edges parallel to the axes of coordinates. Now

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In the figure, the inner (shaded) region $A$ represents a sphere of radius $r_A=1$, within which the electrostatic charge density varies with the radial distance $r$ from the center as ${\rho }_A=kr$, where $k$ is positive. In the spherical shell $B$ of outer radius $r_B$, the electrostatic charge density varies as ${\rho }_B=\frac{2k}{r}$. Assume that dimensions are taken care of. All physical quantities are in their SI units.

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Which of the following statement(s) is/(are) correct?

An insulated sphere with dielectric constant $K$ (where $K>1$) of radius of $R$ is having a total charge $+Q_0$ uniformly distributed in the volume. It is kept inside a metallic sphere of inner radius $2R$ and outer radius $3R$. Whole system is kept inside a metallic shell of radius $4R$, metallic sphere is earthed as shown in the figure. Spherical shell of radius $4R$ is given a charge $+Q_0$. Consider $E-r$ graph for $r>0$ only. Where $E$ and $r$ represents electric field and radial distance from the centre respectively. Then choose the CORRECT option(s)

Three uniformly charged infinite wires with linear charge density $\lambda$ are placed along $x, y$ and $z$ axis respectively. Find the flux of electric field through Gaussian surface given by $x^2+y^2+z^2=1:x>0;y>0;z>0$ If your answer is $\frac{n\lambda }{12{ϵ}_0}$  fill the value of $n$

 The electric field due to unknown charge distribution is given by $\overrightarrow{E}=\frac{k}{r^2}\exp \left(-4r\right)\hat{r}$, where $k$ is a constant in SI units. The total charge (in coulombs) in overall space is equal to

A spherical non-conducting shell of uniform charge density $\sigma$ has a small circular hole cut out of it as shown in the figure. What is magnitude of the electric field just inside the sphere, directly below the centre of the circular hole?

A conducting sphere of radius $R$ and charge $Q$ is surrounded by a medium of resistivity $\text{'}\rho \text{'}$ at $t = 0.$ Due to the medium the charge starts decreasing from it. At $t = t_1$ the charge on the sphere is found $\frac{Q}{4}$ then the value of $t_1$ is [Absolute permittivity of the vacuum is ${\epsilon }_0$]

A non-conducting cylinder of radius $a$ is infinitely long. Its volume charge density $\mathrm{\rho }$ varies linearly as the distance from the axis of the cylinder. If $\mathrm{\rho }$ is zero at the axis and is ${\mathrm{\rho }}_{\mathrm{s}}$ on the surface, the electric intensity due to it is :

A charge $Q$ is located at $P\left(0,0, a\right)$. The flux through the shaded region is $\frac{Q}{\eta {ϵ}_0}$.
Find $\eta$.(The region which is shaded is enclosed by lines $y=x,x=a$ and $y=0$ and extends upto $\infty$ )

The electric field in the space is given by $\overrightarrow{E}=E_0\left(x\hat{\mathrm{i}}+y\hat{\mathrm{j}}+z\hat{\mathrm{k}}\right).$ Consider a right circular cylindrical surface whose radius is '$a$' and height ' $h$'. Now choose the correct option(s).

Which of the four particles contribute to the net electric flux through the closed surface?

Which of the four particles contribute to the electric field at point $P$ on the surface?

Use Gauss's Law to determine the electric field at$a.$

The "Gaussian Surface" for an infinitely large charged plate is pillbox of surface area A and length $2a$ centred about the origin. The plate has positive surface charge density $\sigma$.

How much charge is contained in the pillbox?

The electric field due to an infinitely long, straight line of charge with uniform charge density $\lambda$ points straight away from the line and has magnitude $E=\lambda /\left(2\mathrm{\pi }{\epsilon }_{0}r\right)$, where$r$ is the distance from the wire. Calculate the flux of this electric field through a right cylinder of height $h$ and radius $R$, co-axial with the charged line. Repeat the calculation for a cylinder of radius $2 R$.

A hollow charged metal sphere has radius $r$. If the potential difference between its surface and a point at a distance $3r$ from the centre is$V$, then electric field intensity at a distance $3r$ is:

Figure shows a closed surface which intersects a conducting sphere. If a positive charge is placed at the point $P,$ the flux of the electric field through the closed surface

The figure shows a thick metallic sphere. If it is given a charge $+Q,$ then an electric field will be present in the region