Electric field on

equatorial plane

1)


2) Consider a position X on equatorial plane at a distance r from the origin, as shown below ---->


3)Suppose X makes an angle q with the dipole



4)Now the distance from X to  +q  or   -q   is  ----->

 

Now be remember



5)Now the magnitude of electric field at  X due to  +q 



6)Now the magnitude of electric field at X due to  -q 



Therefore we see magnitude of E+q and E-q are  both equal.

7) Now we got two electric field  vectors as shown below




On resolving the two vectors we got




As  

are equal and opposite so they get cancel each other, because magnitude of

E+q and E-q are equal.

 

So net electric field is


                                                

Here negative sign indicates that Enet  is opposite to the direction of dipole as shown below.



Now from (1.1) we get that

Putting the value of E+q we get





If    r≫a





This is the final electric field at X

Where P = 2aq called the dipole moment of the dipole.

 

Be remember:



For a dipole with charges  +q  and   -q  separated by a distance 2a, the dipole moment of the dipole is