Seems like Saturn

At the origin of x y z xyz coordinates there is a solid sphere with radius r r and charge density ρ \rho . Around it, on x y xy plane, there is a ring with radius R R and charge density λ \lambda . If the magnitude of total electric field at P ( 0 , 0 , H ) P(0,0,H) , such that H > r H>r , can be written as ρ r 3 a H 2 ϵ 0 + λ H R b ( R 2 + H 2 ) 3 2 ϵ 0 \dfrac{\rho \cdot r^3}{a\cdot H^2 \cdot \epsilon_{0}}+\dfrac{\lambda \cdot H\cdot R}{b\cdot (R^2+H^2)^{\frac{3}{2}} \cdot \epsilon_{0}} Find the value of a + b a+b


The answer is 5.

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1 solution

Using the principle of superposition of electric fields we have: E P = E s p h e r e + E r i n g \vec{E}_{P}=\vec{E}_{sphere}+\vec{E}_{ring} Taking as gaussian surface a sphere with radius H H for using Gauss's law : E s p h e r e d s = Q e n c l o s e d ϵ 0 \oint E_{sphere} ds = \frac{Q_{enclosed}}{\epsilon_{0}} E s p h e r e 4 π H 2 = ρ 4 3 π r 3 ϵ 0 \Rightarrow E_{sphere} \cdot 4\pi H^2 = \frac{\rho \cdot \frac{4}{3} \pi r^3}{\epsilon_{0}} E s p h e r e = ρ r 3 3 H 2 ϵ 0 E s p h e r e = ρ r 3 3 H 2 ϵ 0 k ^ \Rightarrow E_{sphere} = \frac{\rho \cdot r^3}{3H^2 \epsilon_{0}} \Rightarrow \vec{E}_{sphere}=\frac{\rho \cdot r^3}{3H^2 \epsilon_{0}}\hat{k} And by using the definition of electric field: E r i n g = 1 4 π ϵ 0 0 2 π λ R d θ ( R 2 + H 2 ) 3 2 ( R c o s ( θ ) , R s i n ( θ ) , H ) \vec{E}_{ring}=\frac{1}{4\pi \epsilon_{0}} \displaystyle \int_{0}^{2\pi} \frac{\lambda \cdot R d\theta}{(R^2+H^2)^{\frac{3}{2}}} (-R cos(\theta), -R sin(\theta), H) By the geometry of the ring, the electric field produced by it in x x and y y directions will be cancelled: E r i n g = λ H 4 π ϵ 0 ( R 2 + H 2 ) 3 2 0 2 π d θ k ^ \vec{E}_{ring}=\frac{\lambda\cdot H}{4\pi \epsilon_{0}(R^2+H^2)^{\frac{3}{2}}}\displaystyle \int_{0}^{2\pi}d\theta \hat{k} E r i n g = λ H 2 ( R 2 + H 2 ) 3 2 ϵ 0 \vec{E}_{ring}=\frac{\lambda \cdot H}{2(R^2+H^2)^{\frac{3}{2}} \cdot \epsilon_{0}} So, as both of them has same direction, the magnitude will be: E P = ρ r 3 3 H 2 ϵ 0 + λ R H 2 ( R 2 + H 2 ) 3 2 ϵ 0 E_{P}=\frac{\rho \cdot r^3}{3\cdot H^2 \cdot \epsilon_{0}}+\frac{\lambda \cdot R\cdot H}{2(R^2+H^2)^{\frac{3}{2}} \cdot \epsilon_{0}}

You must write that point is outside the sphere i.e. H>r.

Archit Agrawal - 4 years, 11 months ago

I think you have got the spelling of "radius" wrong everywhere.

Snehal Shekatkar - 4 years, 11 months ago

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thank you xD

Hjalmar Orellana Soto - 4 years, 11 months ago

Same solution, but knew the results before hand..

Vignesh S - 4 years, 11 months ago

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