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

  sequence  is a continuous and distinctive band of  stars  that appears on plots of stellar  color  versus  brightness. these color-magnitude plots are known as  hertzsprung–russell diagramsafter their co-developers,  ejnar hertzsprungand  henry norris russell. stars on this band are known as  main-sequence stars  or  dwarf stars. these are the most numerous true stars in the universe, and include the earth's  sun.after condensation and ignition of a star, it generates  thermal energy  in its dense core region through  nuclear fusion  of  hydrogen  into  helium. during this stage of the star's lifetime, it is located on the main sequence at a position determined primarily by its mass, but also based upon its chemical composition and age. the cores of main-sequence stars are in  hydrostatic equilibrium, where outward thermal pressure from the hot core is balanced by the inward pressure of  gravitational collapse  from the overlying layers. the strong dependence of the rate of energy generation on temperature and pressure to sustain this balance. energy generated at the core makes its way to the surface and is radiated away at the  photosphere. the energy is carried by either  radiation  or  convection, with the latter occurring in regions with steeper temperature gradients, higher opacity or both.the main sequence is sometimes divided into upper and lower parts, based on the dominant process that a star uses to generate energy. stars below about 1.5 times the  mass of the sun  (1.5  m☉) primarily fuse hydrogen atoms together in a series of stages to form helium, a sequence called the  proton–proton chain. above this mass, in the upper main sequence, the nuclear fusion process mainly uses atoms of  carbon,  nitrogen  and  oxygen  as intermediaries in the  cno cycle  that produces helium from hydrogen atoms. main-sequence stars with more than two solar masses undergo convection in their core regions, which acts to stir up the newly created helium and maintain the proportion of fuel needed for fusion to occur. below this mass, stars have cores that are entirely radiative with convective zones near the surface. with decreasing stellar mass, the proportion of the star forming a convective envelope steadily increases. main-sequence stars below 0.4  m☉undergo convection throughout their mass. when core convection does not occur, a helium-rich core develops surrounded by an outer layer of hydrogen.in general, the more massive a star is, the shorter its lifespan on the main sequence. after the hydrogen fuel at the core has been consumed, the star  evolves  away from the main sequence on the hr diagram, into a  supergiant,  red giant, or directly to a  white dwarf.