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
is the volume of such a shell?
Use Gauss's law to find an expression for the magnitude of the electric field strength E inside the sphere, r≤R, in terms of Q and R.
Express your answer in terms of the variables R, r, Q, and ϵ0.
Homework Answers
For the last r and R are not the same, therefore they are seperate variables. The correct answer is Q/4piE0Rr
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A sphere of radius R has total charge Q. The
volume charge density (C/m3) within the...
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 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)...
(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...
part 1 of 3 Consider a solid insulating sphere of radius b with nonuniform charge density...
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 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 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...
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/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.
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...
Question A1 (12 marks] A sphere with radius R carries a charge density that is proportional...
Question A1 (12 marks] A sphere with radius R carries a charge density that is proportional to the square of the distance from the origin, i.e. p = kr2 for some constant k. (a) [3 marks] Calculate k if the total charge on the sphere is Q. (Hint: dt = r2 sin(O) dr do do ) (b) [3 marks) Write down Gauss's law in integral form. In which situations can it be used to directly calculate the electric field of...
A uniform volume charge occupies the region r < a and has volume charge density ρ....
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...
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...
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 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.
Question A1 (12 marks] A sphere with radius R carries a charge density that is proportional to the square of the distance from the origin, i.e. p = kr2 for some constant k. (a) [3 marks] Calculate k if the total charge on the sphere is Q. (Hint: dt = r2 sin(O) dr do do ) (b) [3 marks) Write down Gauss's law in integral form. In which situations can it be used to directly calculate the electric field of...
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