H C VERMA Solutions for Class 12-science Physics Chapter 8 - Gauss' Law

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Chapter 8 - Gauss' Law Exercise 141

Question 1

The electric field in a region is given by H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law Find the flux of this field through a rectangular surface of area 0.2m2 parallel to the Y - Z plane.

Solution 1

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law (As given plane is parallel to YZ plane)

And flux is given as :

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 2

A charge Q is uniformly distributed over a rod of length l. Consider a hypothetical cube of edge l with the centre of the cube at one end of the rod. Find the minimum possible flux of the electric field through the entire surface of the cube. 

Solution 2

Linear charge density,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

And charge enclose inside is given as:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 3

Show that there can be no net charge in a region in which the electric field is uniform at all points.

Solution 3

Electric field is uniform and given plane is perpendicular to it. Thus it is an equipotential surface with no net current on that surface. So, net charge is :

Q = 0

Question 4

The electric field in a region is given by H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law Find the charge contained inside a cubical volume bounded by the surfaces x=0, x=a, y=a, z=0 and z=a. Take E0=5×103N/c, l=2cm and a=1cm 

Solution 4

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

IfH-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Also,H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 5

A charge Q is placed at the centre of a cube. Find the flux of the electric field through the six surfaces of the cube.

Solution 5

By Gauss Law:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

At centre of cube:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law (For six surfaces)

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 6

A charge Q is placed at a distance a/2 above the centre of a horizontal, square surface of edge 𝛼 as shown in figure. Find the flux of the electric field through the square surface.

 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law

 

Solution 6

Let us assume a cubical surface of side 'a'. So charge will be at centre of the cube and flux through one surface of cube is given as:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 7

Find the flux of the electric field through a spherical surface of radius R due to a charge of 10-7C at the centre and another equal charge at a point 2R away from the centre.

 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law

 

Solution 7

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

 

By gauss's law sphere experiences flux only by charge present inside sphere and not by charge present outside sphere.

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 8

A charge Q is placed at the centre of an imaginary hemispherical surface. Using symmetry arguments and the Gauss's law, find the flux of the electric field due to this charge through the surface of the hemisphere.

 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Solution 8

In case of sphere flux is given as:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

In case of hemisphere,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 9

A spherical volume contains a uniformly distributed charge of density.2.0×10-4C/m3H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law Find the electric field at a point inside the volume at a distance 4.0cm from the centre.

Solution 9

Volume charge density = H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

We know,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Also,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 10

The radius of a gold nucleus (Z = 79) is about H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law Assume that the positive charge is distributed uniformly throughout the nuclear volume. Find the strength of the electric field at (a) the surface of the nucleus and (b) at the middle point of a radius. Remembering that gold is a conductor, is it justified to assume that the positive charge is uniformly distributed over the entire volume of the nucleus and does not come to the outer surface?

Solution 10

(a) We know,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

(b) Now,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Also,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 11

A charge Q is distributed uniformly within the material of a hollow sphere of inner and outer radii r1and r2H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law Find the electric field at a point P a distance H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law away from the centre for r1 < x < r2 Draw a rough graph showing the electric field as a function of H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law for 0 < x < 2r2.H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Solution 11

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Charge enclosed by sphere of radius H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Lawis :

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

By Gauss law,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Chapter 8 - Gauss' Law Exercise 142

Question 12

A charge Q is placed at the centre of an uncharged, hollow metallic sphere of radius αH-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law (a) Find the surface charge density on the inner surface and on the outer surface. (b) If a charge q is put on the sphere, what would be the surface charge densities on the inner and the outer surfaces? (c) Find the electric field inside the sphere at a distance H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law from the centre in the situations (a) and (b).

Solution 12

By Property of induction charge induced at inner surface H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law and outer surface . +Q

(a) Surface charge density at inner surface H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Surface charge density at outer surface H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

(b) When q is added then there is no effect on inner surface charge density

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

But when q is added outer surface charge density becomes,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

(c) Let us assume an area inside sphere which is at H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law distance from centre of sphere.

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 13

Consider the following very rough model of a beryllium atom. The nucleus has four protons and four neutrons confined to a small volume of radius H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law The two 1s electrons make a spherical charge cloud at an average distance of H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law from the nucleus, whereas the two 2s electrons make another spherical cloud at an average distance of H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law from the nucleus. Find the electric field at (a) a point just inside the 1s cloud and (b) a point just inside the 2s cloud.

Solution 13

(a) Electric field ,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

(b) Total Charge Q = H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Electric field ,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 14

Find the magnitude of the electric field at a point 4cm away from a line charge of density H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Solution 14

We know,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 15

A long cylindrical wire carries a positive charge of linear density H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law An electron revolves around it in a circular path under the influence of the attractive electrostatic force. Find the kinetic energy of the electron. Note that it is independent of the radius.

Solution 15

For cylinder

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Also, Force

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 16

A long cylindrical volume contains a uniformly distributed charge of density p. Find the electric field at a point P inside the cylindrical volume at a distance H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law from its axis.

 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

 

Solution 16

We know,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

(Total charge As nQ = H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law where, H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law is volume charge density)

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 17

A nonconducting sheet of large surface area and thickness d contains uniform charge distribution of density p. Find the electric field at a point P inside the plate, at a distance H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law from the central plane. Draw a qualitative graph of E against x for 0 < x < d.

Solution 17

We know,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 18

A charged particle having a charge of H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law is placed close to a nonconducting plate having a surface charge density H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law Find the force of attraction between the particle and the plate.

Solution 18

We know in case of charge conducting sheet:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

And Force is :

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

=H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 0.45 N

Question 19

One end of a 10cm long silk thread is fixed to a large vertical surface of a charged nonconducting plate and the other end is fastened to a small ball having a mass of 10g and a charge of H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law. In equilibrium, the thread makes an angle of 60° with the vertical. Find the surface charge density on the plate.

Solution 19

According to given figure:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

And

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Also

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

From (1) and (2)

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 20

Consider the situation of the previous problem. (a) Find the tension in the string in equilibrium. (b) Suppose the ball is slightly pushed aside and released. Find the time period of the small oscillations.

Solution 20

(a) In equilibrium,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

(b) For this, acceleration is included. Thus,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 21

Two large conducting plates are placed parallel to each other with a separation of 2.00cm between them. An electron starting from rest near one of the plates reaches the other plate in 2.00 microseconds. Find the surface charge density on the inner surfaces.

Solution 21

We know,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Now, electric field is given as:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

And

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Also

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 22

Two large conducting plates are placed parallel to each other and they carry equal and opposite charges with surface density H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law as shown in figure. Find the electric field (a) at the left of the plates, (b) in between the plates and (c) at the right of the plates.

 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law

 

Solution 22

(a) For any point on left plate, electric flux is not present as it is outside the system. Hence, electric field,

E = 0

(b) For charge present in between plates, it experiences force towards negative plate, which is given as:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

(c) For any point of right plate no electric flux is present as it is outside the system. Hence, electric field is given as:

E = 0

Question 23

Two conducting Plates X and Y, each having large surface area A (on one side), are placed parallel to each other shown in figure. The plate X is given a charge Q whereas the other is neutral. Find (a) the surface charge density at the inner surface of the plate X, (b) the electric field at a point to the left of the plates, (c) the electric field at a point in between the plates and (d) the electric field at a point to the right of the plates.

 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

 

Solution 23

(a) Electric field at left end H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

And Electric field at right end H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

As the electric field is balanced

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

And H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

(b) From above part,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law(for left part)

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

As H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law is charged plate H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law electric field is directed towards left.

(c) Here, H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law is charged plate so it acts as only source of electric field. Thus, electric field is directed from left to right and is given as:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

(d) Similar to above part, Y acts as positive plate thus it repels, so electric field is directed towards right and is given as:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Question 24

Three identical metal plates with large surface areas are kept parallel to each other as shown in figure. The leftmost plate is given a charge Q, the rightmost a charge -2Q and the middle one remains neutral. Find the charge appearing on the outer surface of the Rightmost plate.

 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law

 

Solution 24

Net electric field at point P is:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

Solving we get,

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

And, Net charge on right of right plate is:

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law 

 

H-c-verma Solutions Cbse Class 12-science Physics Chapter - Gauss Law