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Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

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Q. 35. Find the electric field strength vector if the potential of this field has the form φ = ar, where a is a constant vector, and r is the radius vector of a point of the field. 

Solution. 35. In accordance with the problem  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus from the equation Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEIrodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 36. Determine the electric field strength vector if the potential of this field depends on x, y coordinates as
 (a) φ = a (x2  — y2);
 (b) φ = axy, 

where a is a constant. Draw the approximate shape of these fields .using lines of force (in the x, y plane).

Solution. 36. Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

So,  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The sought shape of field lines is as shown in the figure (a) of answersheet assuming a > 0:

(b) Since φ = axy 

So,    Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Plot as shown in the figure (b) of answersheet


Q. 37. The potential of a certain electrostatic field has the form cp = a (x2 + y2) + bz2, where a and b are constants. Find the magnitude and direction of the electric field strength vector. What shape have the equipotential surfaces in the following cases: 

(a) a > 0, b> 0; (b) a > 0, b < 0? 

Solution. 37. Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

So,  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Hence  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Shape of the equipotential surface :

Put  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Then the equipotential surface has the equation  

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE and the equation of the equipotential surface is

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

which is an ellipse in p , z coordinates. In three dimensions the surface is an ellipsoid of revolution with semi- axis  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE then the equation is

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

This is a single cavity hyperboloid of revolution about z axis. If φ = 0 then 

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

or   Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

is the equation of a right circular cone. 

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

or  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

This is a two cavity hyperboloid of revolution about z-axis.


Q. 38. A charge q is uniformly distributed over the volume of a sphere of radius R. Assuming the permittivity to be equal to unity throughout, find the potential
 (a) at the centre of the sphere;
 (b) inside the sphere as a function of the distance r from its centre. 

Solution. 38. From Gauss’ theorem intensity at a point, inside the sphere at a distance r from the centre is given by,  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE and outside it, is given by Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(a) Potential at the centre of the sphere,

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEIrodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

as  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEIrodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) Now, potential at any point, inside the sphere, at a distance r from it s centre.

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

On integration :  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 39. Demonstrate that the potential of the field generated by a dipole with the electric moment p (Fig. 3.4) may be represented as Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE where r is the radius vector. Using this expression, find the magnitude of the electric field strength vector as a function of r  and θ. 

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Solution. 39. Let two charges +q and - q be separated by a distance l. Then electric potential at a point at distance r > > l from this dipole,

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE     (1)

But  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

From Eqs. (1) and (2),

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

where p is magnitude of electric moment vector.

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 40. A point dipole with an electric moment p oriented in the positive direction of the z axis is located at the origin of coordinates. Find the projections Ez  and E of the electric field strength vector (on the plane perpendicular to the z axis at the point S (see Fig. 3.4)). At which points is E perpendicular to p? 

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Solution. 40. From the results, obtained in the previous problem,

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

From the given figure, it is clear that,

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

and  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

When  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

So  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE the points located on the lateral surface of the cone, having its axis, coinciding with the direction of z-axis and semi vertex angle Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 41. A point electric dipole with a moment p is placed in the external uniform electric field whose strength equals E0, with Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEIn this case one of the equipotential surfaces enclosing the dipole forms a sphere. Find the radius of this sphere.

Solution. 41. Let us assume that the dipole is at the centre of the one equipotential surface which is spherical (Fig.)* On an equipotential surface the net electric field strength along the tangent of it becomes zero. Thus

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Alternate : Potential at the point, near the dipole is given by,

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

For φ to be constant,

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 42. Two thin parallel threads carry a uniform charge with linear densities λ and —λ. The distance between the threads is equal to l. Find the potential of the electric field and the magnitude of its strength vector at the distance r ≫ l at the angle θ to the vector l (Fig. 3.5).

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Solution. 42. Let P be a point, at distace r >> l and at an angle to θ the vector  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Also,  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 43. Two coaxial rings, each of radius R, made of thin wire are separated by a small distance l (l ≪ R) and carry the charges q and —q. Find the electric field potential and strength at the axis of the system as a function of the x coordinate (Fig. 3.6). Show in the same drawing the approximate plots of the functions obtained. Investigate these functions at |x| ≫ R.

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Solution. 43. The potential can be calculated by superposition. Choose the plane of the upper ring as x = l/2 and that of the lower ring as x = - l/2.

Then   Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEIrodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

For  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The electric field is  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEIrodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

For  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE The plot is as given in the book.


Q. 44. Two infinite planes separated by a distance l carry a uniform surface charge of densities σ and —σ (Fig. 3.7). The planes have round coaxial holes of radius R, with 1 ≪ R. Taking the origin O and the x coordinate axis as shown in the figure, find the potential of the electric field and the projection of its strength vector E on the axes of the system as functions of the x coordinate. Draw the approximate plot φ (x).

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Solution. 44. The field of a pair of oppositely chaiged sheets with holes can by superposition be reduced to that of a pair of uniform opposite charged sheets and discs with opposite charges. Now the charged sheets do not contribute any field outside them. Thus using the result of the previous problem

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The plot is as shown in the answersheet.


Q. 45. An electric capacitor consists of thin round parallel plates, each of radius R, separated by a distance l (l ≪ R) and uniformly charged with surface densities σ and —σ. Find the potential of the electric field and the magnitude of its strength vector at the axes of the capacitor as functions of a distance x from the plates if x ≫ l. Investigate the obtained expressions at x ≫ R. 

Solution. 45. For x > 0 we can use the result as given above and write

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

for the solution that vanishes at α. There is a discontinuity in potential for |x| = 0. The solution for negative x is obtained by σ → -σ. Thus

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Hence ignoring the jump

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

for large  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEIrodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 46. A dipole with an electric moment p is located at a distance r from a long thread charged uniformly with a linear density λ. Find the force F acting on the dipole if the vector p is oriented
 (a) along the thread;
 (b) along the radius vector r;
 (c) at right angles to the thread and the radius vector r. 

Solution. 46. Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE not change as the point of observation is moved along the thread.

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 47. Find the interaction force between two water molecules separated by a distance l = 10 nm if their electric moments are oriented along the same straight line. The moment of each molecule equals p = 0.62.10-29  C • m. 

Solution. 47. Force on a dipole of moment p is given by,

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

In our problem, field, due to a dipole at a distance l, where a dipole is placed,

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Hence, the force of interaction,

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 48. Find the potential φ (x, y) of an electrostatic field E = a (yi xj), where a is a constant, i and j are the unit vectors of the x and y axes. 

Solution. 48. 

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

On integrating,  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 49. Find the potential φ (x, y) of an electrostatic field E = 2axyi + a (x2  — y2) j, where a is a constant, i and j are the unit vectors of the x and y axes. 

Solution. 49.  

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

or   Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

On integrating, we get,

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 50. Determine the potential φ (x, y, z) of an electrostatic field E = ayi (ax + bz) j + byk, where a and b are constants, i, j, k are the unit vectors of the axes x, y, z. 

Solution. 50. Given, again

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

On integrating,

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 51. The field potential in a certain region of space depends only on the x coordinate as φ = — ax3 + b, where a and b are constants. Find the distribution of the space charge p (x).

Solution. 51. Field intensity along x-axis.

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE     (1)

Then using Gauss’s theorem in differential from 

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 52. A uniformly distributed space charge fills up the space between two large parallel plates separated by a distance d. The potential difference between the plates is equal to Δφ. At what value of charge density p is the field strength in the vicinity of one of the plates equal to zero? What will then be the field strength near the other plate? 

Solution. 52. In the space between the plates we have the Poisson equation

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

or,  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

where p0 is the constant space charge density between the plates. 

We can choose  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Then  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEIrodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

NowIrodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

if    Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

then Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Also Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE


Q. 53. The field potential inside a charged ball depends only on the distance from its centre as φ = ar2 + b, where a and b are constants. Find the space charge distribution p (r) inside the ball.

Solution. 53. Field intensity is along radial line and is

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE                      (1)

From the Gauss’ theorem, 

Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

where dq is the charge contained between the sphere of radii r and r + dr. 

Hence  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE  (2)

Differentiating  Irodov Solutions: Constant Electric Field in Vacuum - 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

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