u/No_Student2900

Can you help me see why that angle I have highlighted is 4θ given that θ is as shown in Figure 6? I've been trying to follow this solution and I've understood pretty much everything aside from how they got that angle as 4θ. For example it's easy to see why 4θ=90° because of the central angle theorem for circles.

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Can you help me see why the width of the rings from the spherical surface and infinite sheet is given by ldθ/sinα=ldθ/cosθ? I do get why the two α's and two θ's in the figure are the same (for the θ part it can be explained by invoking the alternate interior angle). Geometric construction is not my strong suit so I'm really struggling trying to understand why the width of the rings are given as such.

Can you help me see why the width of the rings from the spherical surface and infinite sheet is given by ldθ/sinα=ldθ/cosθ? I do get why the two α's and two θ's in the figure are the same (for the θ part it can be explained by invoking the alternate interior angle). Geometric construction is not my strong suit so I'm really struggling trying to understand why the width of the rings are given as such.

For z values below the base of the hemispherical shell wouldn't the necessary figure be like that shown in the 4th image? This will then give us r²=R² + z² +2Rzcosθ instead of r²= R² + z²-2Rzcosθ, affecting the denominator in Equation 12.27. I could follow the solutions manual but I'm really skeptical if we could apply the results in Eq. 12.29 for points along the negative z axis. I could also convince myself that the triangle in Figure 12.10 would be correct for points along the positive z-axis (but not sure if it is still correct for points along the negative z-axis). What are your thoughts on this one?

In my attempt of this problem I simplified the expression (r²+R²-2rRcosθ)^(1/2) into (R²-2rRcosθ)^(1/2) since I figured r<<R. I then factored it out to R(1-2(r/R)cosθ)^(1/2), giving me the expression U(r)=2 ∫ from 0 to π of qλ/4πε₀(1-2(r/R)cosθ)^(1/2) dθ, which if we apply the Taylor expansion with ε=-2(r/R)cosθ will give us U(r)=2 ∫ from 0 to π of qλ/4πε₀ *(1+ rcosθ/R + 3r²cos²θ/2R²) dθ. Applying the integral we then finally get U(r)=qλ/4πε₀ (2π + 3r²π/2R²) and dU/dr=3qλr/4πε₀R² and if we use this to construct the Newton's second law equation the ω we will get is 3 times larger than the ω shown in the solutions. So I wanna ask if it was wrong to simplify (r² + R² -2rRcosθ)^(1/2) into (R²-2rRcosθ)^(1/2) ? Can you tell me when and where is it generally safe to invoke such approximations to simplify the expressions?

The solution said plugging in the limits of -1000 to 1000 into the double summation will give -0.115 but when I tried it myself in Mathematica I got -0.133 and the two numbers are significantly different. Is there something wrong with my syntax that it gave off a different answer? Which of these two numbers is the correct one?

Also, would the math be still the same if I let the double summation run from n=1 to ∞ and m=1 to ∞ and then multiply the result by 2, and then add that to the result of the single summation from m=1 to ∞ multiplied by 2?

Title says it all, can you recommend me songs that sound like these? I don't really know how to describe exactly what is it about these songs that make them attractive to me. If it helps I got hook sa Heaven Is a Place on Earth after watching the Black Mirror episode San Junipero.