Question

A thin non conducting ring of radius r is a linear charge density lambda equals to lambda not cos theta well and are not is the value of lambda at theta equals to zero find net electric dipole moment for this charge distribution

Answer 1

We know,
Electric dipole is the product of speration between poles and chare on each pole. I mean, P = R × q
Here, R is speration between two poles and q is charge on each pole .
For elementary part ,
$\bold{P = \int{R.dq}}$
In case of ring , R = rcosθ i + rsinθ j , where r is radius of ring.
and dq = λ.dl
We know , charge is the product of linear charge density and Length .
Here, Given, λ = λ₀cosθ and dl = rdθ
So, dq = rλ₀cosθ.dθ

Now, P = $\bold{P = \int\limits^{2\pi}_0{(r cos\theta i +r sin\theta j).(r\lambda_0cos\theta.d\theta)}}$
so, Px = $\bold{P = \int\limits^{2\pi}_0{r^2\lambda_0 cos^2\theta}\,d\theta}}}$
After solving you will get ,
Px = r²λ₀π

Similarly , Py = $\bold{P = \int\limits^{2\pi}_0{r^2\lambda_0 cos\theta.sin\theta}\,d\theta}}}$
Solve this one , you will get
Py = 0
Hence, P = Px i + Py j
= r²λ₀π i + 0j
= r²λ₀π i

Hence, electirc dipole moment = r²λ₀π along x axis .