How to Solve a Gauss' Law Problem in a Spherical Region?

AI Thread Summary
To solve the Gauss' Law problem in a spherical region, the radial electric field can be derived from the voltage function v(r) = wr^p, resulting in E(r) = -Pwr^(p-1). Gauss' Law states that the electric flux through a closed surface equals the enclosed charge divided by the permittivity of free space (Q/ε0). To find the charge enclosed within a sphere of radius r, the relationship between the electric field and charge density must be established using the derived electric field. The discussion highlights confusion regarding the application of Gauss' Law to the voltage function, indicating a need for clarification on relating electric fields to charge distributions. Understanding these concepts is essential for completing the remaining parts of the problem.
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Homework Statement



In a spherical region, the voltage is measured to be spherically symmetrical, with v=v(r)=wr^p
a. Find the radial electric field.
b. Use Gauss’ Law to find the charge enclosed in a sphere of radius r.
c. Find the charge enclosed by a sphere of radius r+dr.
d. Find the differential charge enclosed in the annular region between two concentric spheres of radii r and r+dr.
e. Find the differential volume of the annular region between two concentric spheres of radii r and r+dr.
f. Find the charge density, rho=rho(r)=?


Homework Equations





The Attempt at a Solution


i am pretty sure that part a would be v(r)= - integral E(r) dr. so the radial electric field would be -Pwr^(p-1)

i am confused on how to do the rest. any help/hints would be appreciated.
 
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Well what does gauss' law state?
 
Gauss' law states that the flux of the electric field is equal to Q/epsilon0.

my question then is how do i relate that to the given statement v(r)=wr^p?
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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