(1 pt) suppose f is a radial force field, s1 is a sphere of radius 7 centered at the origin, and the flux integral ∫∫s1f⋅ds=3. let s2 be a sphere of radius 63 centered at the origin, and consider the flux integral ∫∫s2f⋅ds. (a) if the magnitude of f is inversely proportional to the square of the distance from the origin, what is the value of ∫∫s2f⋅ds?

(a) ∫∫s₂ f⋅ds = 12

(b) ∫∫s₂ f⋅ds = 3/4

Step-by-step explanation:

From the question we were given s₁ to be = radius of the sphere R₁ = 7

and the Radius of the other sphere S₂ = R₂ = 63

the Flux integral, ∫∫s₁ f⋅ds  = 3

(a). we are asked to fine the value of ∫∫s₂ f⋅ds

but first we know that the surface of the sphere = 4πR²

given f is inversely proportional to the square of the distance from the origin;

f ∝ 1/R²

f = K/R      where k is a proportionality constant

but recall that surface of the sphere = 4πR² = 196π

∴ ∫∫s₁ f⋅ds = 3 = f(4πR²) = 196πf = 3 (1)

equating gets,

196π(K/R²) = 3 = 196π (K/7₂)

  3 = K = 3/4π

considering the second sphere s₂ of radius R₂;

∫∫s₂ f⋅ds = (K/R²)(4π R²) = 4π (3/π ) = 12

∫∫s₂ f⋅ds = 12

(b). already we know, ds ∝ R²

given that f ∝ 1/R³

thus we have  ∫∫s f⋅ds =  ∫∫s 1/R³*R² =  ∫∫s 1/R

if R increases by a factor = 16/4 = R₂/R₁, we have 4

so our surface integral reduces by 1/4

s₂ = s₁/4

s₂ = 3/4

our answer becomes ∫∫s₂ f⋅ds = 3/4

cheers i hope this helps

"a" determine whether sage found the correct quotient. show your work

step-by-step explanation: