Learning Goal: To use Gauss' Law with a non-uniform charge distribution. A nonuniform, but spherically symmetric, insulating sphere of charge has a charge density p(r) given as follows: P(r) Po(1-4r/3R) P(T) = 0 for r≤R for r > R where p is a positive constant. This means that the amount of charge per unit volume varies with distance from the center of the sphere. Part A Find the total charge contained in the insulating sphere. To find this, use the integral expression given in the problem introduction. Express your answer in terms of the variables r, R, po, and appropriate constants. ΜΕ ΑΣΦ Q= 2 P(r) dV where dV = 4πr² dr (in other words, you Submit Request Answer Hint: The charge enclosed in a given spherical shell of inner radius R₁ and outer radius R2 is Qenc have to integrate over each spherical shell of radius r from the inner radius to the outer radius). Notice that if the charge is uniform, so that p is a constant, and you have a solid sphere with inner radius zero and outer radius R (i.e. a normal sphere) you can pull p out of the integral and it reduces to the simple expression Qenc PπR³. For a non-uniformly charged sphere, however, as in this problem, you have to use the full integral to find the charge enclosed in a given region. Part B ? Obtain an expression for the electric field in the region r > R. Express your answer in terms of the variables r, R, po, and appropriate constants. ▸ View Available Hint(s) ΜΕ ΑΣΦ ? E₁ = Submit Constants Learning Goal: To use Gauss' Law with a non-uniform charge distribution. A nonuniform, but spherically symmetric, insulating sphere of charge has a charge density p(r) given as follows: P(r) Po(1-4r/3R) P(r) = 0 for r < R for r > R where p is a positive constant. This means that the amount of charge per unit volume varies with distance from the center of the sphere. Hint: The charge enclosed in a given spherical shell of inner radius R₁ and outer radius R₂ is R₂ Qene = P(r) dV where dV = 4πr² dr (in other words, you have to integrate over each spherical shell of radius r from the inner radius to the outer radius). Notice that if the charge is uniform, so that p is a constant, and you have a solid sphere with inner radius zero and outer radius R (i.e. a normal sphere) you can pull p out of the integral and it reduces to the simple expression Qenc =PπR³. For a non-uniformly charged sphere, however, as in this problem, you have to use the full integral to find the charge enclosed in a given region. Part C Obtain an expression for the electric field in the region r < R. Express your answer in terms of the variables r, R, Po, and appropriate constants. ▸ View Available Hint(s) E₂ = Submit Part D ΜΕ ΑΣΦ Find the value of rat which the electric field is maximum. ? Express your answer in terms of the variables r, R, Po, and appropriate constants. ▸ View Available Hint(s) Η ΜΕ ΑΣΦ ? T= Submit Constants Learning Goal: To use Gauss' Law with a non-uniform charge distribution. A nonuniform, but spherically symmetric, insulating sphere of charge has a charge density p(r) given as follows: P(r) Po(1-4r/3R) P(r) = 0 for r≤R for r > R where p is a positive constant. This means that the amount of charge per unit volume varies with distance from the center of the sphere. Hint: The charge enclosed in a given spherical shell of inner radius R₁ and outer radius R₂ is R₂ Qenc = P(r) dV where dV = 4πr² dr (in other words, you have to integrate over each spherical shell of radius r from the inner radius to the outer radius). Notice that if the charge is uniform, so that p is a constant, and you have a solid sphere with inner radius zero and outer radius R (i.e. a normal sphere) you can pull p out of the integral and it reduces to the simple expression Qenc = PR³. For a non-uniformly charged sphere, however, as in this problem, you have to use the full integral to find the charge enclosed in a given region. Part D Find the value of r at which the electric field is maximum. Express your answer in terms of the variables r, R, Po, and appropriate constants. ▸ View Available Hint(s) T= Submit Part E ΜΕ ΑΣΦ Find the value of that maximum field. ? Express your answer in terms of the variables r, R, Po, and appropriate constants. Emax = ΜΕ ΑΣΦ ? Submit Request Answer Constants