A positive charge distribution exists within a nonconducting spherical region of radius a. The volume charge density rho is not uniform but varies with the distance r from the center of the spherical charge distribution, according to the relationship rho = br for 0 < r < a, where b is a positive constant, and rho = 0, for r > a. a. In terms of B, a, and fundamental constants, determine the total charge Q in the spherical region.

b. In terms of B, r, a, and fundamental constants, determine the magnitude of the electric field at a point a distance r from the center of the spherical charge distribution for each of the following cases.
i. r > a
ii. r = a
iii. 0 < r < a

c. Quantitatively graph the electric field vs. r for 0 < r , 3a. Include labels and values on the axes.

d. In terms of B, a, and fundamental constants, determine the electric potential at a point a distance r from the center of the spherical charge distribution for each of the following cases:
i. r > a
ii. 0 < r < a

a)  Q = π b a⁴ , b) E = k π b a⁴ / r² , E = kπ b a² ,  E = k π b r² ,

d) V = - k π b a⁴ / r , V = - k π b r³ / 3

Explanation:

a) the charge density is defined by

     ρ = dQ / dV

     dQ = ρ dV

the volume differential is the area of the sphere by the change in radius

     dV = (4π r²) dr

  we integrate to enter the charge

     Q = ∫  ρ (4π r²) dr

we replace and integrate

      Q = 4π b ∫ r r² dr

      Q = 4π b r⁴ / 4

we evaluate between the integration limits, lower r = 0 and higher r = a

      Q = π b a⁴  

We define a Gaussian surface in spherical shape, in this case the field line is parallel to the radii of the sphere, so the scalar product is reduced to the ordinary product

         The area of a sphere is

         A = 4π r²

         E 4π r² = qint /ε₀

         E = qint / (4πε₀) 1 / r²

         k = 1 / 4π ε₀

we must find the charge inside the gaussian surface

b) Calculate the electric field for different points, for this we use Gauss's law

i) r> a

      Ф = E . dA = qint /ε₀

    qint = π b a4

whereby the electric field is

         E = k π b a⁴ / r²

ii) r = a

         E = kπ b a²

iii) 0 <r <a

in this case the charge inside the gaussian surface is

         qint = π b r⁴

  the electric field is

         E = k π b r⁴ / r²

         E = k π b r²

c) see attached

d) The electric potential is

      ΔV = - ∫ E .dr

we have two regions

i) r> a

    ΔV = - ∫ (k pi b a⁴ / r²) dr

    ΔV = - k π b a⁴  ∫ dr / r²

   ΔV = - k π b a⁴ (1 / r)

we evaluate for the lower limit r = r and the upper limit r = ∞, we make the potential zero for infinite distance

     V = - k π b a⁴ / r

ii) in the region

     0 <r <a

   ΔV = - ∫ (k π b r²) dr

   ΔV = -k π b r³ / 3

in this case we make that for r = 0 the potential is zero

    V = - k π b r³ / 3

sleep paralysis is a parasomnia, events associated with sleep.

sleep paralysis happens when you are in the middle of waking up or falling asleep. it is the feeling of being conscious but unable to move. it definitely involves hallucinations.

i had a friend that had sleep paralysis and he told me a story one time where he was laying in bed and he couldn't move his body no matter how hard he tried and there was someone standing in the corner of his room coming towards him. he couldn't move and you can't "wake up" because you are already conscious. sometimes sleep paralysis can be triggered by frequent waking up during the night based off of multiple trips to the bathroom, or if you had ever been abused, sleep paralysis can start happening from things like that.

heat the gas and measure the pressure