Evaluate the integrals (a) (3-2r-1)-3)dr (b) Write an expression for the volume charge density p(r) of...
A sphere of radius R has total charge Q. The volume charge density (C/m3) within the sphere is ρ(r)=C/r2, where C is a constant to be determined. The charge within a small volume dV is dq=ρdV. The integral of ρdV over the entire volume of the sphere is the total charge Q. Use this fact to determine the constant C in terms of Q and R. Hint: Let dV be a spherical shell of radius r and thickness dr. What...
PROBLEM 2: A thick, spherical, insulating shell has an inner radius a and an outer radius b. The region a< r < b has a volume charge density given by p = A/r where A is a positive constant. At the center of the shell is a point charge of electric charge +q Determine the value of A such that the electric field magnitude, in the region a < r < b, is constant.
consider a spherical ball of of charge radius R with a volume charge density of p(r)=a^3 for r≤R what are coefficient unit, calculate the electrical field r≥R and show that the expression agrees when r=R
part 1 of 3 Consider a solid insulating sphere of radius b with nonuniform charge density p = ar, where a is a constant. Find the charge contained within the radius r<b as in the figure. The volume element dV for a spherical shell of radius r and thickness dr is equal to 47 r2 dr. part 2 of 3 If a = 5 x 10-6 C/m' and b = 1 m, find E at r = 0.6 m. The...
(a) A solid sphere, made of an insulating material, has a volume charge density of p , where r is the radius from the center of the sphere, a is constant, and a >0. What is the electric field within the sphere as a function of the radius r? Note: The volume element dv for a spherical shell of radius r and thickness dr is equal to 4tr2dr. (Use the following as necessary: a, r, and co.) magnitude E direction...
PROBLEM4 (a) Write an expression for the mass density ρ(r) of a point particle with a mass m at a position r; How much is the volume integral of ρ over the entire space? (b) What is the mass density of a system of two point particles with mass m each. One of the particle is at the origin of the coordinate system and the other one is at ä. How much is the integral of the density over the...
6. In this problem you will learn how to use Dirac delta functions to solve integrals and define densities of point charges. (a) Using the definition of Dirac delta function, evaluate the following integrals 15) 产00 (i) (4x2-8x-1) δ(x-4) dx (ii) sin x δ(x-π/2) dx x3 δ(x + 3)dx In(x + 3)δ(x + 2)dx (b) What is the volume charge density of an electric dipole, consisting of a point charge -q at (c) What is the integral of this charge...
4.1 A sphere of radius R has a uniform volume charge density ρ(r) Pr. A. Calculate E(r) B. Use your answer to A to calculate V(r). C. Use your answer to B to calculate the energy of this charge configuration, via the expression U pV d where the integral must be evaluated over the bounded charge distribution. D. Use your answer to A to calculate the energy of this charge configuration, via the expression 2 2 space
A uniform volume charge occupies the region r < a and has volume charge density ρ. The total charge of the volume charge is +2Q. A thin spherical shell of radius b > a surrounds the volume charge. The total charge of the thin spherical shell is −2Q. (a)Draw the electric field map with 4 lines per Q on the figure to the right. (b)Calculate the electric field everywhere. This part is to be worked symbolically (no decimal numbers should...
3. A non-conducting sphere (R-0.05 m) is charged with a non-uniform charge density pr)-(0.64 a)-~ 0.2037:r (in units of Cim3). For a variable distance rin from the center within the sphere, integrate da p(r)-dV from the center (r 0) out to rin to find the charge qemerin) contained within the radius rin R. [reminder: the differential volume of a thin shell is dV= 4nr2dr Evaluate qen at r,-R to find the total charge Qo on the sphere. ( At a...