A sphere of radius R has total charge Q. The volume charge density (C/m^{3}) within the sphere is \(\rho=\rho_{0}(1-(r^{2}/R^{2}))\)
This charge desity decreases quadratically from \(\rho_{0}\)
b) Show that the electric field inside the sphere points radially outward with magnitude
c) Show that your results of part (b) has the expected value at r=R.
b)
integrating charge density
int rho_0 (1-r^2/R^2) *dV = Q
we get rho_0 = Q*15/(8R^3)--1
by guass law
E*4 pi r^2 = charge enclosed/epsilon_0---2
charge enclosed = int rho * dV
= rho_0 *4 pi (r^3/3 - r^5/(5R^2))--3
substituting 2 and 1 and 3 in 2
we get
E = Qr /(8 pi epsilon_0*R^3)(5-3(r/R)^2)--4
c)
at r=R
charge enclosed = Q
E*4 pi R^2 = Q/epsilon_0
==> E = Q/[4pi R^2*epsilon_0]
we get the same by substituting r =R in 4
A sphere of radius R has total charge Q. The volume charge density (C/m^{3}) within the...
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...
(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...
A solid, insulating sphere of radius a has a uniform charge density throughout its volume and a total charge of Q. Concentric with this sphere is an uncharged, conducting hollow sphere whose inner and outer radii are b and c as shown in the figure below. We wish to understand completely the charges and electric fields at all locations. (Assume Q is positive. Use the following as necessary: Q, ε0 , a, b, c and r. Do not substitute numerical...
An insulating hollow sphere of inner radius R1 and outer radius R2 has a uniform volume charge density pand carries a total positive charge Q. A. Calculate the magnitude of the electric field and the electric flux at a point r where: B. Sketch the electric field and the electric flux as a function of r.
A non-conducting sphere of radius 5.6 cm has a uniform volume charge with a charge density of rho = 41.4 mu C/m^3. What is the electric field at r = 2.95 cm?
1. A very long, uniformly charged cylinder has radius R and charge density \rho. Determine the electric field of this cylinder inside (r<R) and outside (r>R)2. A large, flat, nonconducting surface carries a uniform surface charge density σ. A small circular hole of radius R has been cut in the middle of the sheet. Determine the electric field at a distance z directly above the center of the hole.3. You have a solid, nonconducting sphere that is inside of, and...
Charge Q is distributed uniformly throughout the volume of an insulating sphere of radius R = 4.00 cm. At a distance of r = 8.00 cm from the center of the sphere, the electric field due to the charge distribution has magnitude 640 N/C . a. What is the volume charge density for the sphere? Express your answer to two significant figures and include the appropriate units. b. What is the magnitude of the electric field at a distance...
A solid sphere, made of an insulating material, has a volume charge density of ρ = a/r 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 4πr2dr. (Use the following as necessary: a, r, and ε0.), where r is the radius from the center of the sphere, a is constant, and a > 0. magnitude E= (b)...
A cylindrically symmetric charge distribution has a volume charge density that depends on the radial position, r, as follows: rho(r) = rho0(1-r/a) where rho0 is the charge density at the central axis and "a" is just a size constant. What is the electric field at r = 2a? (For direction, “+” means radially outward and “-” means radially inward.)
Consider a charged sphere of radius R. The charge density is not constant. Rather, it blows up at the center of the sphere, but falls away exponentially fast away from the center, p(r)=(C/r2)e-kr where C is an unkown constant, and k determines how fast the charge density falls off. The total charge on the sphere is Q. a) Write down the Electric Field outside the sphere, where r ≥ R, in term of the total Q. b) Show that C=...