A non-uniformly charged sphere of radius R has a total charge Q. The electric field inside this charge distribution is described by E=Emax(r4 /R4 ), where Emax is a known constant. Using the differential form of Gauss’s law, find volume charge density as a function of r. Express your result in terms of r, R and Emax.
To find the volume charge density (ρ) as a function of r for the non-uniformly charged sphere, we'll start with Gauss's law in differential form, which states:
∇ · E = ρ / ε₀
where ∇ · E is the divergence of the electric field E, ρ is the volume charge density, and ε₀ is the permittivity of free space.
Given the electric field inside the charged sphere as:
E = Emax * (r^4 / R^4)
Now, we need to find the divergence of the electric field (∇ · E). Since the electric field is spherically symmetric, we can write it in terms of the radial coordinate (r) only:
E = E(r) * r̂
where r̂ is the radial unit vector.
The divergence in spherical coordinates is given by:
∇ · E = (1/r^2) * (∂ / ∂r) (r^2 * E(r))
Substituting the expression for E(r) into the divergence formula:
∇ · E = (1/r^2) * (∂ / ∂r) (r^2 * Emax * (r^4 / R^4))
∇ · E = (1/r^2) * Emax * (∂ / ∂r) (r^6 / R^4)
∇ · E = (1/r^2) * Emax * (6r^5 / R^4)
Now, equating this to ρ / ε₀, we get:
(1/r^2) * Emax * (6r^5 / R^4) = ρ / ε₀
Solving for ρ:
ρ = (6ε₀ / Emax) * (r^5 / R^4)
Expressing the result in terms of r, R, and Emax, the volume charge density (ρ) inside the non-uniformly charged sphere is:
ρ(r) = (6ε₀ / Emax) * (r^5 / R^4)
A non-uniformly charged sphere of radius R has a total charge Q. The electric field inside...
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=...
A spherical ball of charge has radius R and total charge Q. The electric field strength inside the ball(r ? R) is E(r)=Emax(r^(4)/R^(4)). 1) What is Emax in terms of Q and R? 2) Find an expression for the volume charge density ?(r) inside the ball as a function of r.
An electric charge Q is distributed uniformly throughout a non-conducting sphere of radius r0, See Fig. below. Using the Gauss's law, determine the electric field: a) Outside of sphere (r0>r). b) Inside the sphere (r0<r).
Consider a uniformly volume‑charged sphere of radius R and charge Q . What is the electric potential on the surface of the sphere in terms of R , Q , and ϵ 0 , choosing the zero reference point for the potential at the center of the sphere?
A sphere has a total charge Q uniformly distributed over its volume. The field inside the sphere at a radius r is given by Er= k (Q/R^3) r (a) What is the electric field at a radius r from the center of the sphere, where r > R (i.e outside of the sphere)? (b) Write down an expression for the electric potential at a radius r for r > R (i.e. outside of the sphere). (c) What is the electric...
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 uniformly charged non-conducting sphere of radius a is placed at the center of a spherical conducting shell of inner radius b and outer radius c. A charge +Q is distributed uniformly throughout the inner sphere. The outer shell has charge -Q. Using Gauss' Law: a) Determine the electric field in the region r< a b) Determine the electric field in the region a < r < b c) Determine the electric field in the region r > c d)...
1) (a) A conducting sphere of radius R has total charge Q, which is distributed uniformly on its surface. Using Gauss's law, find the electric field at a point outside the sphere at a distance r from its center, i.e. with r > R, and also at a point inside the sphere, i.e. with r < R. (b) A charged rod with length L lies along the z-axis from x= 0 to x = L and has linear charge density λ(x)...
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...
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.