Question

A thin non-conducting ring of radius R has a linear charge density λ = λ∘ cosθ ,where λ∘ is the value of λ at θ = 0.Find net electric dipole moment for this charge distribution

Answer 1

Answer:

we know that

By the definition of electric dipole moment we can write

p⃗ =∫r⃗ dqdq =λdlFor ring dl =Rdθ where R is radius of the ringdq =λocosθ R dθ

Here, in formula of electric dipole moment, r is position of charge dq.

Now consider

r⃗ = (R cosθ, R sinθ, 0) That is origin at centre of ring, then

p⃗ =∫2π0 R2λocos θ dθ ( cosθ, sinθ, 0)

px =∫2π0 R2λocos θ cosθ dθ =R2λo∫2π0 cos2 θ dθ

px =R2λo2∫2π0(1+cos 2θ)dθ =R2λo2[θ+12sin2θ]2π0=R2λoπ

py =∫2π0 R2λocos θ sinθ dθ

=R2λo2∫2π0 2cos θ sinθ dθ

py =R2λo2∫2π0 sin2θ dθ =R2λo2[cos2θ]2π0=0

pz =0

Therefore

p⃗ =px=R2λoπ xˆ

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