Obtain the ratio of Electric field formed by an ideal dipole at points lying on axial and equatorial axis.

Asked by avaneesh1162 | 12th Jun, 2021, 11:06: AM

Expert Answer:

Let a dipole consists of charges +q and -q that are separated by a distance d is placed at origin of
X-Y coordinate system so that mid point of dipole coinicides with origin.
 
Electric field at a point x = r on x-axis is given as
 
begin mathsize 14px style E subscript x space equals space E subscript plus space minus space E subscript negative space space end subscript equals space left parenthesis space K space cross times space q space right parenthesis space open parentheses 1 over open parentheses r minus begin display style d over 2 end style close parentheses squared space minus space 1 over open parentheses r space plus begin display style d over 2 end style close parentheses squared close parentheses space end style 

 
begin mathsize 14px style E subscript x space equals space fraction numerator left parenthesis space K space cross times space q space right parenthesis over denominator open parentheses r squared space minus space begin display style d squared over 4 end style close parentheses squared end fraction cross times left parenthesis 2 space r space d right parenthesis end style ..................................(1)

 
Where K = 1/ (4πεo ) is Coulomb's constant
 
if we consider dipole moment p = ( q × d )   and  r >> d/2 , then eqn.(1) is written as
 
Ex  = K × [ ( 2 p ) / r3  ]  ............................ (2)
 
Electric field at a point y = r on y-axis is given as
 
Ey = 2 E cosθ  .................................(3)
 
Where  E is given as
begin mathsize 14px style E space equals space E subscript plus space equals space E subscript minus space equals space K space cross times fraction numerator q over denominator open parentheses r squared space plus space begin display style d squared over 4 end style close parentheses end fraction end style
Hence we get Eas   ,  begin mathsize 14px style E subscript y space equals space 2 space cross times K space cross times fraction numerator q cross times space left parenthesis d divided by 2 right parenthesis over denominator open parentheses r squared space plus space begin display style d squared over 4 end style close parentheses to the power of 3 divided by 2 end exponent end fraction space equals space K cross times p over r cubed end style .........................(4)

In above expression , we used the approximation r >> d  and  dipole moment p = ( q × d ) 
 
From eqn.(2) and eqn.(4) , we get  ( Ex / Ey )  = 2 

Answered by Thiyagarajan K | 12th Jun, 2021, 02:59: PM