At the origin of x y z coordinates there is a solid sphere with radius r and charge density ρ . Around it, on x y plane, there is a ring with radius R and charge density λ . If the magnitude of total electric field at P ( 0 , 0 , H ) , such that H > r , can be written as a ⋅ H 2 ⋅ ϵ 0 ρ ⋅ r 3 + b ⋅ ( R 2 + H 2 ) 2 3 ⋅ ϵ 0 λ ⋅ H ⋅ R Find the value of a + b
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You must write that point is outside the sphere i.e. H>r.
I think you have got the spelling of "radius" wrong everywhere.
Same solution, but knew the results before hand..
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Using the principle of superposition of electric fields we have: E P = E s p h e r e + E r i n g Taking as gaussian surface a sphere with radius H for using Gauss's law : ∮ E s p h e r e d s = ϵ 0 Q e n c l o s e d ⇒ E s p h e r e ⋅ 4 π H 2 = ϵ 0 ρ ⋅ 3 4 π r 3 ⇒ E s p h e r e = 3 H 2 ϵ 0 ρ ⋅ r 3 ⇒ E s p h e r e = 3 H 2 ϵ 0 ρ ⋅ r 3 k ^ And by using the definition of electric field: E r i n g = 4 π ϵ 0 1 ∫ 0 2 π ( R 2 + H 2 ) 2 3 λ ⋅ R d θ ( − R c o s ( θ ) , − R s i n ( θ ) , H ) By the geometry of the ring, the electric field produced by it in x and y directions will be cancelled: E r i n g = 4 π ϵ 0 ( R 2 + H 2 ) 2 3 λ ⋅ H ∫ 0 2 π d θ k ^ E r i n g = 2 ( R 2 + H 2 ) 2 3 ⋅ ϵ 0 λ ⋅ H So, as both of them has same direction, the magnitude will be: E P = 3 ⋅ H 2 ⋅ ϵ 0 ρ ⋅ r 3 + 2 ( R 2 + H 2 ) 2 3 ⋅ ϵ 0 λ ⋅ R ⋅ H