Flux is a key concept related to the Field theory. But what is Flux ? Literally it just ?flow?

Find the flux of the uniform electric field through each of the five surfaces of the inclined plane as shown in figure. What is the total flux through the entire closed surface ?

Gauss Law relates the flux of electric field through a closed surface to the Net charge enclosed by it. But there are a lot of fine points that we need to understand about Gauss Law.

Is Gauss Law applicable to the Field of enclosed charge or to the Net Field in space ? Although most of the books state that it is applicable only to the NET Field in space, but it is actually applicable to BOTH of the above cases.

Which of the following statement is /are correct?

a) Gauss law applies to a closed surface of any shape

b) According to Gauss law, if a closed surface encloses no charge, then the electric field must vanish everywhere on the surface

c) Gauss law is applicable only when there is a symmetrical distribution of charge.

d) Electric flux through a closed surface is equal to total flux due to all the charges enclosed within that surface only.

e) Electric field calculated by Gausss law is the field due to only those charges which are enclosed inside the Gaussian surface.

For Gausss law, mark out the correct statement(s).

a) If we displace the enclosed charges ( within a gaussian surface ) without crossing the boundary, then both E and f remain same.

b) If we displace the enclosed charges without crossing the boundary, then E changes but f remains the same.

c) If charge crosses the boundary, then both E and f would change.

d) If charge crosses the boundary, then f changes but E remains the same.

Under what condition can the electric flux f be found through a closed surface?

a) If the total charge inside the surface is specified.

b) Only if the location of each point charge inside the surface is specified.

c) If the total charge outside the surface is specified.

d) If the magnitude of electric field is known everywhere on the surface.

Figure shows the field produced by two point charges +q and -q of equal magnitude but opposite signs (an electric dipole).

Find the electric flux through each of the closed surface A, B, C and D.

For which surface is the flux highest ?

The electric flux from a cube of edge l is f. If edge of the cube is made 2l and charge enclosed is halved, its value will be

a) 4f

b) 2f

c) f/2

d) f

A Gaussian surface encloses two of the four positively charged particles.

a) which of the four particles contribute to the electric field at point P on the

surface?

b) which of the four particles contribute the net electric flux through the

Gaussian surface?

If Coulombs law involved 1/r3 (instead of 1/r2) would Gausss law still be true ?

Explanation and Derivation of the equation of Solid Angle.

Solid Angle of a Closed Surface at a point inside and outside the surface.

Find the solid angle subtended by a disc of radius R at a point lying at distance d from the center of ring and on its perpendicular axis.

Derivation of Gauss Law from Solid Angle

A point charge q is located on the axis of a disk of radius R at a distance b from the plane of the disc. Find the flux through the disc.

If the flux through the disc is one-sixth of the total flux over a closed surface enclosing the charge, then find the relation between b and R.

Gauss Law and Spherical Symmetry. Derivation of Electric Field due to a uniformly charged Shell and a Uniformly and NON-uniformly charged solid sphere.

Fig. shows two solid spheres with uniformly distributed charge throughout their volumes. Each has radius R. Point P lies on a line connecting the centers of the sphere, at a distance R/2 from the center of sphere 1.

If the net electric field at point P is zero, what is the ratio q2/q1 of the total charge q2 in sphere 2 to the total charge q1 in sphere 1?

Figure shows a non-conducting shell with charge +q distributed uniformly over its surface. A point charge +q is placed outside the shell.

Is the net field inside the shell zero ? Can we conclude this using Gauss Law ?

Draw net field in the space.

Gauss Law and Cylindrical Symmetry. Derivation of Electric Field due to a uniformly charged Cylindrical Shell and a Uniformly charged solid Cylinder.

Figure shows an infinite rod of radius R1 and volume charge density r1 inside an infinite thin non-conducting cylindrical shell of radius R2 and volume charge density r2. Find the electric field

a) inside the rod

b) in the region between the rod and shell

c) outside the shell

Gauss Law and Planar Symmetry. Derivation of Electric Field due to a uniformly charged infinite Sheet and Slab.

Fig. shows three uniformly charged infinite plastic sheets. Graph gives the net electric field along a x-axis in various regions in space. What is the sum of charge density on sheets ? What is the ratio of the charge density on sheets ?

A point charge q is placed at a distance a/2 from the center of a square of side a. Calculate the electric flux passing through the square.

A hemispherical body is placed in a uniform electric field E. What is the flux linked with the curved surface, if the field is

(a) parallel to base of the body

(b) perpendicular to base of the body

A point charge Q is located just above the center of the flat face of a hemisphere of radius R.

What is the flux

a) through the curved surface

b) through the flat face

c) net flux through the hemisphere

Repeat the above questions, if the charge is exactly at the center.

In figure, what is flux through each face of the cube if a charge q is placed, such that

a) its center is exactly on the corner of cube ?

b) charge is just outside the corner of cube ?

A disk of radius a/4 having a uniformly distributed charge 6C is placed in the x-y plane with its centre at ( - a/2, 0, 0 ). A rod of length carrying a uniformly distributed charge 8C is placed on the x-axis drom x = a/4 to x = 5a/4. Two point charges - 7C and 3C are placed at ( - a/4, - a/4, 0 ) and ( - 3a/4, 3a/4, 0 ), respectively. Consider a cubical surface formed by six surfaces

x = a/2, y = a/2, z = a/2. The electric flux through this cubical surfaces is

A cubical region of side l has its centre at the origin. It encloses three fixed point charges, -q, +3q and ?q placed symmetrically on y-axis.

Choose the correct options.

(a) the net electric flux crossing the plane x = +l/2 is equal to the net electric flux crossing the plane x = -l/2

(b) the net electric flux crossing the plane y = +l/2 is more than the net electric flux crossing the plane y = -l/2

(c) the net electric flux crossing the plane z = +l/2 is equal to the net electric flux crossing the plane z = -l/2

(d) the net electric flux crossing the plane x = +l/2 is equal to the net electric flux crossing the plane y = -l/2

e) the net electric flux through the cube is q/eo

Figure shows four equal charges placed on the corners of a square. A cubical surface is drawn as shown. What is the net flux through the cube ?

Field lines only seem to enter the surface, so how is the net flux zero ?

Find the flux through sphere as a function of d, for the 3 cases shown below.

a) Sphere of radius R at distance d from infinite line of charge

b) Sphere of radius R at distance d from infinite sheet of charge

c) Sphere of radius R at distance d from the center of ring of radius R

Take the charge density to constant for all cases.

A charged particle is placed at the center of two thin non-conducting concentric shells as shown in the figure. Graph gives the net flux f through a Gaussian sphere centered on the particle, as a function of the radius r of the sphere.

a) What is the charge on central particle?

b) What are the net charges of shell A and B ?

c) Plot the variation of electric field in the regions

d) How would the field change if the point charge is removed ?

A hollow dielectric sphere, as shown in Fig. has inner and outer radii of R1 and R2, respectively. The total charge carried by the sphere is +Q.

Then,

a) can we say that the electric field for r < R1 is zero irrespective of whether the charge is distributed uniformly or not ?

Given that the charge is uniformly distributed between R1 and R2, find

b) the electric field for R1 < r < R2

c) the electric field for r > R2

A sphere of radius R has a uniform volume charge density r. A spherical cavity of radius r whose centre lies at distance a from the center of sphere, is removed from the sphere. a) Find the electric field at any point inside the spherical cavity. b) Find the electric field outside the cavity but inside the sphere. c) Find the electric field outside the sphere.

A uniformly charged solid spherical region has two cavities of equal size as shown.

Field inside each cavity is uniform True / False ?

An infinitely long solid cylinder of radius R has a uniform volume charge density r. It has a spherical cavity of radius R/2 with its centre on the axis of the cylinder, as shown in the fig. The magnitude of the electric field at the point P, which is at a

distance 2R from the axis of the cylinder, is given by the expression . The value of a is ?

A cube of side l has one corner at the origin of coordinates and extends along the positive x-,y- and z- axes. Suppose the electric field in this region is given by E = (a + by) j. Determine the charge inside the cube. a and b are some constants.

A uniformly charged rod of length L moves towards left with a small but constant speed v. At t = 0, the left end of the rod just touches the right face of an imaginary cube of side L/2.

Which of the graphs represents the flux of the electric field through the cube as the rod goes through it ?

An electron is placed just in the middle between two long fixed line charges of charge density + l each. The wires are in the xy plane. (Ignore gravity).

(a) The equilibrium of the electron will be unstable along x direction

(b) The equilibrium of the electron will be neutral along y direction

(c) The equilibrium of the electron will be stable along z direction

(d) The equilibrium of the electron will be stable along y direction

Two thin non-conducting spherical shells are fixed in place on a x axis. Shell 1 has radius R1 and charge q1 distributed uniformly on its outer surface. Shell 2 has radius R2 and charge - q/h distributed uniformly on its outer surface. Their centers are separated by distance l. Other than at x = , where on the x axis is the net electric field equal to zero?

A very long solid insulating cylinder with radius R has a cylindrical hole with radius d bored along its entire length. The axis of the hole is a distance r from the axis of the cylinder, where r < d < R. The solid material of the cylinder has a uniform volume charge density r.

a) Find the magnitude and direction of the electric field inside the hole, and show that this is uniform over the entire hole.

b) What if the cavity is spherical, with radius = d with its center at distance r from the axis of cylinder. (r < d < R)

A small circular hole of radius R has been cut in the middle of an infinite, non-conducting surface that has uniform charge density s.

What is the electric field at point P at a distance z from the center of hole.

Two oppositely charged spheres of radius R overlap such that the distance between their centers is d.

Show that the field in the overlapping region is constant.

What is the value of field ?

A solid spherical region with radius R, has a spherical cavity whose diameter is equal to the radius of the spherical region. Total charge on the remaining portion is Q.

Find the electric field at a point P as shown.

The electric field in a region is radially outward with magnitude E = ar.

Calculate the charge contained in a sphere of radius r centred at the origin.

Find vol charge density.

A system consists of a ball of radius R carrying a uniformly distributed charge q and surrounding space filled with a charge of volume density r = a/r where a is a constant and r is distance from the centre of the ball. Find the charge on the ball for which the magnitude of electric field strength outside the ball is constant ? The dielectric constant of ball and surrounding may be taken equal to unity.

A solid sphere of radius R has a charge Q distributed in its volume with a charge density r = kra, where k and a are constants and r is the distance from its centre. If the electric field at r = R/2 is 1/8 times that at r = R, Find the value of a.

A solid insulating sphere of radius R has a non-uniform charge density that varies with r according to the expression r = ar2, where a is a constant and r < R is measured from the center of the sphere. Show that

(a) the magnitude of the electric outside (r > R) the sphere is E = a R5 / 5 eo r2

(b) the magnitude of the electric field inside (r < R), the sphere is E = a r3 / 5 eo

Two point charges q and - q are separated by a distance 2a. Evaluate the flux of electric field through a disc of radius R lying in the middle of two charges with its centre on the line joining the charges.

Find the flux through an infinite plane midway between the charges.

Find the flux through a disc of radius R placed at the end of a semi-infinite wire of charge. Axis of disc coincides with the wire and the disc is perpendicular to the wire. The liner charge density on the wire is l.

Find the electric flux crossing the rectangular sheet of length l, width b and whose centre is at a distance d from an infinite line of charge with linear charge density l. Consider that the plane of frame is perpendicular to the infinite line of charge.

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