The linear charge density on the circumference of a ring of radius 'R' is $λ = λ_{0} sin θ$ where $θ$ is defined as shown in figure. Find the magnitude of electric dipole moment (in coulomb.m) of the ring if $λ_{0} = \frac{1}{π}$ SI unit and $R = 1 m$ .

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TestHub · Accepted Answer

P res ​ = ∫ ( d P ) sin θ = ∫ ( d q 2 R ) sin θ = 2 R ∫ ( λ d ℓ ) sin θ = 2 R λ 0 ​ ∫ ( sin θ d ℓ ) sin θ = 2 R λ 0 ​ ∫ ( sin 2 θR d θ ) = 2 R 2 λ 0 ​ ∫ 0 π ​ sin 2 θ d θ = 2 R 2 λ 0 ​ ⋅ 2 π ​ = π R 2 λ 0 ​ = π × 1 × π 1 ​ = 1 C.m

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