A solid insulating sphere of radius R has a non-uniform charge density ρ = Ar2 ,...
A solid insulating sphere of radius R has a non-uniform charge density ρ = Ar2 , where A is a constant and r is measured from the center of the sphere.
a) Show that the electric field outside the sphere (r > R) is E = AR5 /(5εor 2 ).
b) Show that the electric field inside the sphere (r < R) is E = AR3 /(5εo).
Hint: The total charge Q on the sphere is found by integrating ρ dV, over appropriate limits inside or outside the sphere. Also, dV can be treated as very thin, spherical shells where dV = 4πr 2 dr.
Please be as clear and detailed as possible, explaining each step
Homework Answers
for second part consider a
Gausian surface which is a sphere of radius r , concentric with
given charged sphere. Then Gauss' Law is applied.
We have to take the charge enclosed within the Gaussian surface ( from r=0 to r = r)
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A solid insulating sphere of radius R has a non-uniform
charge density ρ = Ar2 ,...
A solid sphere, made of an insulating material, has a volume charge density of ρ =...
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)...
3rd Question Consider a solid insulating sphere of radius b with nonuniform charge density ρ-ar, where...
3rd Question
Consider a solid insulating sphere of radius b with nonuniform charge density ρ-ar, where a is a constant. Find the charge contained within the radius r< bas in the figure. The volume element dV for a spherical shell of radius r and thickness dr is equal to 4 π r2 dr.
A solid, insulating sphere of radius a has a uniform charge density ρ and a total charge Q
Guided Problem 4 -Gauss's LawA solid, insulating sphere of radius a has a uniform charge density ρ and a total charge Q. Concentric with this sphere is an uncharged, conducting hollow sphere whose inner and outer radii are b and c as shown in the following figure. (a) Find the magnitude of the electric field in the regions: r<a, a<r<b, and r>c. (b) Determine the induced charge per unit area on the inner and outer surfaces of the hollow sphere.Solution scheme:...
(a) A solid sphere, made of an insulating material, has a volume charge density of p...
(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...
Consider a charged sphere with the following charge density ρ(r) =(ρ0(1− r Rmax) r ≤ Rmax...
Consider a charged sphere with the following charge density ρ(r)
=(ρ0(1− r Rmax) r ≤ Rmax 0 r > Rmax
Using Gauß’ law, calculate the electric field
(a) ~ E1 inside the sphere (i.e. r ≤ Rmax),
(b) ~ E2 outside the sphere (i.e r ≥ Rmax),
(c) Check that lim r→Rmax ~ E1 = lim r→Rmax ~ E2. Reminder: Due
to spherical symmetryRRRV ρ(r0)dxdydz =Rr 0 ρ(r0)4πr02dr0
Please provide an explanation for the
solution.
Problem 5. Consider a charged...
Charge is distributed throughout a spherical volume of radius R with a density ρ ar where...
Charge is distributed throughout a spherical volume of radius R with a density ρ ar where α is a constant. an risthe distance from the center of the sphere. Determine the electric field due to the charge at a point a distance r from the center that is inside the sphere, and at a point a distance r from the center that is outside the sphere. (Enter the radial component of the electric field. Use the following as necessary: R,...
A sphere of radius R has total charge Q. The volume charge density (C/m3) 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 solid, insulating sphere of radius a has a uniform charge density of P and a total charge of Q.
A solid, insulating sphere of radius a has a uniform charge density of P and a total charge of Q. Concentric with this sphere is a conducting spherical shell with inner and outer radii are b and c, and having a net charge -3Q. (a) (5 pts.)Use Gauss's law to derive an expression for the electric field as a function of r in the regions r < a (b) (4 pts.) Use Gauss's law to derive an expression for the electric field...
A hollow insulating sphere of inner radius "a" and outer radius "b" has a non-uniform charge...
A hollow insulating sphere of inner radius "a" and outer radius "b" has a non-uniform charge per unit volume p that varies with distance r from the center of the sphere according to the expression p=Cr^2, where C is a given constant. a) what is the total charge Q contained in the hollow sphere b) what is the electric field at a point inside the sphere, a< r < b
Find the electric field due to a charged insulating sphere (radius R) with non-uniform charge density...
Find the electric field due to a charged insulating sphere
(radius R) with non-uniform charge density rho=beta*r^2 with
beta>0.
Find the electric field due to a charged insulating sphere (radius R) with non-uniform charge density rho=beta*r^2 with beta greaterthan 0.
3rd Question
Consider a solid insulating sphere of radius b with nonuniform charge density ρ-ar, where a is a constant. Find the charge contained within the radius r< bas in the figure. The volume element dV for a spherical shell of radius r and thickness dr is equal to 4 π r2 dr.
(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...
Consider a charged sphere with the following charge density ρ(r)
=(ρ0(1− r Rmax) r ≤ Rmax 0 r > Rmax
Using Gauß’ law, calculate the electric field
(a) ~ E1 inside the sphere (i.e. r ≤ Rmax),
(b) ~ E2 outside the sphere (i.e r ≥ Rmax),
(c) Check that lim r→Rmax ~ E1 = lim r→Rmax ~ E2. Reminder: Due
to spherical symmetryRRRV ρ(r0)dxdydz =Rr 0 ρ(r0)4πr02dr0
Please provide an explanation for the
solution.
Problem 5. Consider a charged...
Charge is distributed throughout a spherical volume of radius R with a density ρ ar where α is a constant. an risthe distance from the center of the sphere. Determine the electric field due to the charge at a point a distance r from the center that is inside the sphere, and at a point a distance r from the center that is outside the sphere. (Enter the radial component of the electric field. Use the following as necessary: R,...
Find the electric field due to a charged insulating sphere
(radius R) with non-uniform charge density rho=beta*r^2 with
beta>0.
Find the electric field due to a charged insulating sphere (radius R) with non-uniform charge density rho=beta*r^2 with beta greaterthan 0.
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