Electric Field due to Infinite Line Charge

Electric Field due to Infinite Line Charge

AU : Hay-06, 07, 12, 15, 16, 17, 18, Dec.-03, 06, 08, 09, 14

• Consider an infinitely long straight line carrying uniform line charge having density ρ_L C/m. Let this line lies along z-axis from -∞ to ∞ and hence called infinite line charge. Let point P is on y-axis at which electric field intensity is to be determined. The distance of point P from the origin is 'r' as shown in the Fig. 2.6.1.

• Consider a small differential length dZ carrying a charge dQ, along the line as shown in the Fig. 2.6.1. It is along z-axis hence dZ = dz.

dQ = ρ_L dl = ρ_L dz ... (2.6.1)

• The co-ordinates of dQ are (0, 0, z) while the co-ordinates of point P are (0, r, 0). Hence the distance vector  can be written as,

Note : For every charge on positive z-axis there is equal charge present on negative z-axis. Hence the z component of electric field intensities produced by such charges at point P will cancel each other. Hence effectively there will not be any z component of   at P. This is shown in the Fig. 2.6.2.

Note : For such an integration, use the substitution

z = r tan θ i.e. r = z/tan θ

dz = r  sec² θ dθ

• Here r is not the variable of integration.

 For z = -∞ , θ = tan^-1 (-∞) = - π/2 = -90^o

For z = +∞, θ = tan^-1 (∞) = π/ 2 = + 90°

Key Point : If without considering symmetry of charges and without cancelling z component from d  , if integration is carried out, it gives the same answer. The integration results the z component of   to be mathematically zero.

• The result of equation (2.6.5) which is specifically in cartesian system can be generalized. The  is unit vector along the distance r which is perpendicular distance of point P from the line charge. Thus in general 

• Hence the result of   can be expressed as,

 where r = Perpendicular distance of point P from the line charge

  = Unit vector in the direction of the perpendicular distance of point P from the line charge

Very important notes : 1. The field intensity  at any point has no component in the direction parallel to the line along which the charge is located and the charge is infinite. For example if line charge is parallel to z-axis,  can not have  component, if line charge is parallel to y-axis,   cannot have  component. This makes the integration calculations easy.

2. The above equation consists r and  which do not have meanings of cylindrical co-ordinate system. The distance r is to be obtained by distance formula while  is unit vector in the direction of .

Key Point : This result can be used as a standard result for solving other problems.

Ex. 2.6.1 A uniform line charge ρ_L = 25 nClm lies on the line x = -3 m and y = 4m in free space. Find the electric field intensity at a point (2, 3, 15) m. AU : May-06, Marks 8

Sol. : The line is shown in the Fig. 2.6.3. The line with x = - 3 constant and y = 4 constant is a line parallel to z axis as z can take any value. The  at P (2, 3, 15) is to be calculated.

The charge is infinite line charge hence  can be obtained by standard result,

To find , consider two points, one on the line which is (-3, 4, z) while P (2, 3, 15). But as line is parallel to z axis,  cannot have component in direction hence z need not be considered while calculating .

Ex. 2.6.2 An infinitely long uniform line charge is located at y = 3, z = 5. If ρ_L =30 nC/m, find the field intensity  at i) Origin ii) P(0, 6, 1) iii) P(5, 6, 1).AU: Dec.-08, Marks 8

Sol. : The charge is shown in the Fig. 2.6.4.

The charge is parallel to x-axis hence  cannot have any component in x direction hence do not consider x while calculating 

As  does not have any component in x direction and y, z, co-ordinates are same as in (ii) hence  also remains same as obtained in (ii).

Ex. 2.6.3 Obtain an expression for electric field intensity due to a uniformly charged line of length 'l'.AU : May-15, Marks 8

Sol. : The charge of length l is shown in the Fig. 2.6.5 along x-axis.

 Let us obtain expression for  at P(0, 0, d) located on z-axis. Consider differential length dZ = dx along the charge.

As charge is symmetrical along x axis, there can not be x component of  Hence ax part in equation (1), need not be considered.

Examples for practices

Ex. 2.6.4 A uniform line charge, infinite in extent with ρ_L = 20 nC/m lies along the z-axis. Find the  at (6, 8, 3) m. 

Ex. 2.6.5 Two identical uniform line charges, with ρ_L = 75 nC/m are located in free space at x = 0, y = ± 0.4 m. What force per unit length does each line charge exert on the other ?

[Ans.: 0.7899 µN]

Ex. 2.6.6 A line charge density ρ_Lis uniformly distributed over a length of 2a with centre as origin along x-axis. Find  at a point P which is on the z-axis at a distance d.

Ex. 2.6.7 Two uniform line charges of density ρ_l = 4 nC/m lie in the x = 0 plane at y = ± 4 m. Find E at (4, 0, 10) m. 

Review Question

1. Derive an expression for electric field intensity  due to an uniformly charged infinitely long straight line with constant charge density in C/m. 

AU : May-07, Marks 10, Dec.-03, 06, 09, 14, Marks 16, 17,18, Marks 8