Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

I. E. Irodov Solutions for Physics Class 11 & Class 12

JEE : Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

The document Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev is a part of the JEE Course I. E. Irodov Solutions for Physics Class 11 & Class 12.
All you need of JEE at this link: JEE

Q. 86. Half the space between two concentric electrodes of a spherical capacitor is filled, as shown in Fig. 3.14, with uniform isotropic dielectric with permittivity ε. The charge of the capacitor is q. Find the magnitude of the electric field strength between the electrodes as a function of distance r from the curvature centre of the electrodes.

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Solution. 86. At the interface of the dielectric and vacuum,

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

The electric field must be radial and

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev


Q. 87. Two small identical balls carrying the charges of the same sign are suspended from the same point by insulating threads of equal length. When the surrounding space was filled with kerosene the divergence angle between the threads remained constant. What is the density of the material of which the balls are made?

Solution. 87. In air the forces are as shown. In K-oil,

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Since the inclinations do not change

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

or,  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

or,  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

where p0 is the density of K-oil and p that of the material of which the balls are made.


Q. 88. A uniform electric field of strength E = 100 V/m is generated inside a ball made of uniform isotropic dielectric with permittivity ε = 5.00. The radius of the ball is R = 3.0 cm. Find the maximum surface density of the bound charges and the total bound charge of one sign. 

Solution. 88. Within the ball the electric field can be resolved into normal and tangential components.

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

(Since we are interested in the total charge of one sign we must intergrate cos θ from 0 to 1 only).


Q. 89. A point charge q is located in vacuum at a distance l from the plane surface of a uniform isotropic dielectric filling up all the half-space. The permittivity of the dielectric equals a. Find:
 (a) the surface density of the bound charges as a function of distance r from the point charge q; analyse the obtained result at 1→0;
 (b) the total bound charge on the surface of the dielectric.

Solution. 89. The chaige is at A in the medium 1 and has an image point at A’ in the medium 2. The electric field in the medium 1 is due to the actual charge q at A and the image charge q' at A'. The electric field in 2 is due to a corrected charge c[ at A. Thus on the boundary between 1 and 2,

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev
Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev
Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev
The boundary conditions are

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

So,  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

(a) The surface density of the bound charge on the surface of the dielectric

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

(b) Total bound charge   Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev


Q. 90. Making use of the formulation and the solution of the foregoing problem, find the magnitude of the force exerted by the charges bound on the surface of the dielectric on the point charge q.

Solution. 90. The force on the point chaige q is due to the bound charges. This can be calculated from the field at this chaige after extracting out the self field. This image field is

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Thus,  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev


Q. 91. A point charge q is located on the plane dividing vacuum and infinite uniform isotropic dielectric with permittivity ε. Find the moduli of the vectors D and E as well as the potential q  as functions of distance r from the charge q.

Solution. 91. 

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev
Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

 

Thus, 

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev
Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev
Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev


Q. 92. A small conducting ball carrying a charge q is located in a uniform isotropic dielectric with permittivity ε at a distance l from an infinite boundary plane between the dielectric and vacuum. Find the surface density of the bound charges on the boundary plane as a function of distance r from the ball. Analyse the obtained result for l→ 0. 

Solution. 92.  

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev
Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Using the boundary conditions,

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

This implies

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Then, as earlier,

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev


Q. 93. A half-space filled with uniform isotropic dielectric with permittivity a has the conducting boundary plane. Inside the dielectric, at a distance l from this plane, there is a small metal ball possessing a charge q. Find the surface density of the bound charges at the boundary plane as a function of distance r from the ball. 

Solution. 93. To calculate the electric field, first we note that an image charge will be needed to ensure that the electric field on the metal boundary is normal to the surface.

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

The image chaige must have magnitude Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

This is the density of bound charge on the surface.


Q. 94. A plate of thickness h made of uniform statically polarized dielectric is placed inside a capacitor whose parallel plates are interconnected by a conductor. The polarization of the dielectric is equal to P (Fig. 3.15). The separation between the capacitor plates is d. Find the strength and induction vectors for the electric field both inside and outside the plates.

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Solution. 94. Since the condenser plates are connected,

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev
Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev


Q. 95. A long round dielectric cylinder is polarized so that the vector P = αr, where α is a positive constant and r is the distance from the axis. Find the space density p' of bound charges as a function of distance r from the axis.

Solution. 95. Given  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev distance from the axis. The space density of charges is given by, p' = - div  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

On using. 

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev


Q. 96. A dielectric ball is polarized uniformly and statically. Its polarization equals P. Taking into account that a ball polarized in this way may be represented as a result of a small shift of all positive charges of the dielectric relative to all negative charges,
 (a) find the electric field strength E inside the ball;
 (b) demonstrate that the field outside the ball is that of a dipole located at the centre of the ball, the potential of that field being equal to φ = p0r/4π ε0, where P0 is the electric moment of the ball, and r is the distance from its centre.

Solution. 96. In a uniformly charged sphere, 

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

The total electric field is

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

where  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev (dipole moment is defined with its direction being from the -vc chaige to +ve charge.)

The potential outside is 

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRevIrodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev is the total dipole moment.


Q. 97. Utilizing the solution of the foregoing problem, find the electric field strength E0 in a spherical cavity in an infinite statically polarized uniform dielectric if the dielectric's polarization is P, and far from the cavity the field strength is E.

Solution. 97. The electric field  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev in a spherical cavity hi a uniform dielectric of permittivity e is related to the far away field   Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev in the following mannenjmagine the cavity to be filled wp with the dielectric. Then there will be a uniform Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev everywhere and a polarization  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev by, 

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Now take out the sphere making the cavity, the_£lectric field inside the sphere will be 

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev


Q. 98. A uniform dielectric ball is placed in a uniform electric field of strength E0. Under these conditions the dielectric becomes polarized uniformly. Find the electric field strength E inside the ball and the polarization P of the dielectric whose permittivity equals e. Make use of the result obtained in Problem Q.96. 

Solution. 98. By superposition the field  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev the bll is given by

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

On the other hand, if the sphere is not too small, the macroscopic equation  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Also Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev


Q. 99. An infinitely long round dielectric cylinder is polarized uniformly and statically, the polarization P being perpendicular to the axis of the cylinder. Find the electric field strength E inside the dielectric. 

Solution. 99. This is to be handled by the same trick as in Q.96. We have effectively a two dimensional situation. For a uniform cylinder full of charge with chaige density p0 (charge per unit volume), the electric field E at an inside point is along the (cylindrical) radius vector Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev and equal to,

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Therefore the polarized cylinder can be thought of as two equal and opposite charge distributions displaced with respect to each other

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Since  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev (direction of electric dipole moment vector being from the negative charge to positive charge .)


Q. 100. A long round cylinder made of uniform dielectric is placed in a uniform electric field of strength E0. The axis of the cylinder is perpendicular to vector E0. Under these conditions the dielectric becomes polarized uniformly. Making use of the result obtained in the foregoing problem, find the electric field strength E in the cylinder and the polarization P of the dielectric whose permittivity is equal to ε. 

Solution. 100.  As in Q.98, we write   Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

using here the result of the foregoing problem.  

Also   Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

So,  Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRevIrodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

Related Searches

practice quizzes

,

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

,

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

,

ppt

,

video lectures

,

Sample Paper

,

Viva Questions

,

Previous Year Questions with Solutions

,

Important questions

,

Irodov Solutions: Conductors And Dielectrics In An Electric Field- 3 Notes | EduRev

,

Semester Notes

,

study material

,

Free

,

past year papers

,

Exam

,

mock tests for examination

,

MCQs

,

Objective type Questions

,

Extra Questions

,

Summary

,

pdf

,

shortcuts and tricks

;