In my book dipole moment is defined for two charges.
But I got a hw question to calculate dipole moment due to a ring.
I tried two different methods to solve this problem but I don't think that any of them is correct.
I sticked to the definition given in my book for 2 charges,"An electric dipole is a pair of equal and opposite point charges separated by a distance 2a."
Small charge $dq$ = $λRdθ$.
Now by geometry distance between any two points, $2a$ = $2Rcosθ$ as shown in the image.
Now $dp$ = $2adq$
Integrated it from $-π/2$ to $π/2$ and got option A as the answer.
Net Dipole moment will clearly be in the $+x$ direction
Now second method,
Consider any two diametrically opposite points, by symmetry dipole moment will be in the $+x$ direction as it will be cancelled in the $y$ direction.
Now since they are diametrically opposite so $2a$ = $2R$
Hence, $dp$ = $2Rdqcosθ$ in the $+x$ direction.
But I don't think any of my method is correct as each charge will be under the influence of all the opposite charges so I can't actually decide what will $2a$ be.
If the points are diametrically opposite then it will be $2R$ but the distance can be anything as we can choose any two pair of opposite charge so the distance between the opposite charges can be anything from $0$ to $2R$.