A single sphere of water is created by joining 1000 of water spheres of radius r and charge (+Q).If the potential of the small and big spheres are w and v respectively, find v/w.
My thoughts :
The masses of the spheres are proportional to the third power of there radius.
So
1000*(r^3)=R^3
R=10r(R-radius of the big sphere)
V=1/(4π*epsilon) (Q/r)
So v is proportional to (Q/r)
So can we simply take the charge of the big sphere as 1000Q,as it is created by 1000 small spheres being joined?
Yes. Consider Gauss' Law: the charge of each of those small spheres can be considered to be a charge Q at the center, even if was distributed throughout the volume.
So can we consider the bigger sphere has a 1000Q charge at its center?
yes. (of course, you have to do that only if the volumetric charge is distributed evenly), law Gauss Law.
To find the ratio v/w, we need to determine the potential of the small spheres (w) and the potential of the big sphere (v).
The total charge of the big sphere is indeed the sum of the charges of the small spheres. Since each small sphere has a charge of +Q, the total charge of the big sphere is 1000Q.
However, we can't directly assume that the potential of the big sphere is equal to 1000 times the potential of a single small sphere. The potential depends on the size and shape of the distribution of charge, and simply adding up the potentials of individual charges won't give an accurate result.
To find the potential of the small spheres (w), we can use the formula:
w = 1/(4πε₀) * Q/r
where ε₀ is the vacuum permittivity, Q is the charge (+Q) of each small sphere, and r is the radius of each small sphere.
To find the potential of the big sphere (v), we need to relate the radius of the big sphere (R) to the radius of the small sphere (r). Since the big sphere is created by joining 1000 small spheres, the volume of the big sphere must be equal to the sum of the volumes of the small spheres:
1000 * (4/3)π(R³) = 1000 * (4/3)π(r³)
Simplifying this equation, we get:
R³ = 1000r³
Taking the cube root of both sides gives:
R = 10r
Now, we can calculate the potential of the big sphere (v) using the formula:
v = 1/(4πε₀) * (1000Q/R)
Substituting R = 10r, we get:
v = 1/(4πε₀) * (1000Q/(10r))
Simplifying, we have:
v = 1/(4πε₀) * (100Q/r)
Now that we have the values for w and v, we can calculate the ratio v/w:
v/w = [(1/(4πε₀) * (100Q/r)) / (1/(4πε₀) * (Q/r))] = (100Q/r) / (Q/r) = (100Q/r) * (r/Q) = 100
Therefore, the ratio v/w is equal to 100.