An infinitely long, solid insulating cylinder with radius a has positive charge uniformly distributed throughout it with a constant charge per unit volume p.
a) using Gauss's law, derive the expression for the electric field inside the cylinder r<a from the axis of the cylinder in terms of the charge density p.
b) now when r>a
c) Explain how your results show that the electric field created by the solid cylinder is identical to that of the infinite line of charge for points outside of the cylinder. What is the relationship between the cylinders volume density p and the line density lambda.
I will be happy to check your work, I am uncertain where you are faltering on this.
I am uncertain on how to use gauss's law to derive the expression for the electric fields
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html
study that. Sally, flux is Electric field * surface area which is equal to totalenclosedcharge/epislon
so when you make a gaussian surface like a cylinder of length L and radius r...
E*L*PI*2r=totalenclosed charge/epislon
and total enclosed charge (for a spaced charge)= volumeenclosed*charge density.
Now outside the charged region, the total enclosed charge is just volume of the region( L*PI*a^2*charge density) and IT DOES NOT INCREASE.
a) To derive the expression for the electric field inside the cylinder (r < a), we can use Gauss's law. Gauss's law states that the flux of the electric field through a closed surface is directly proportional to the charge enclosed by the surface.
Let's consider a Gaussian surface in the form of a cylindrical surface with radius r and length L (we can assume L > a to ensure that the entire cylinder is enclosed). The cylinder's symmetry suggests that the electric field is constant in magnitude and directed radially inward/outward. Hence, the electric field's direction is along the axis of the cylinder.
\phi_E = E * 2πrL, where \phi_E is the electric flux through the cylinder.
The charge enclosed by the Gaussian surface is given by the volume charge density multiplied by the volume enclosed:
Q_enclosed = p * πr^2L
Applying Gauss's law, we have:
\phi_E = Q_enclosed / ε_0
E * 2πrL = p * πr^2L / ε_0
Simplifying, we get:
E = p * r / (2ε_0)
Therefore, the expression for the electric field inside the cylinder (r < a) is E = p * r / (2ε_0).
b) When r > a, the electric field can be determined by considering the charge enclosed by a Gaussian surface with radius r. Since the cylinder is infinitely long, the charge enclosed by this surface is the same as the charge enclosed by a cylindrical volume of radius a and height L, where L is greater than a.
Q_enclosed = p * πa^2L
Applying Gauss's law again, we have:
\phi_E = Q_enclosed / ε_0
E * 2πrL = p * πa^2L / ε_0
Simplifying, we get:
E = (p * a^2) / (2ε_0r)
Therefore, for r > a, the expression for the electric field is E = (p * a^2) / (2ε_0r).
c) The results from parts (a) and (b) show that the electric field created by the solid cylinder is identical to that of an infinite line of charge for points outside the cylinder. For r > a, the expression for the electric field depends only on the charge density p, the radius of the cylinder a, and the distance from the axis r.
Now, let's consider the relationship between the volume density p and the line density λ.
The line density λ is defined as the charge per unit length along the axis of the cylinder. We can imagine slicing the cylinder into circular rings of thickness Δr and length L. The charge contained in each ring is given by ΔQ = p * π(r^2Δr). The length of each ring is L.
The total charge contained in the cylinder is given by integrating the charge density over the entire length L:
Q_total = ∫ p * π(r^2Δr) dr (integrated from 0 to a)
= p * π ∫(r^2 dr) (integrated from 0 to a)
= p * π (a^3/3)
Therefore, the line density λ is equal to p * π (a^3/3) / L. But since the cylinder is infinitely long, we can think of L as effectively becoming infinite, making the line density λ = p * π (a^3/3) / ∞.
Since λ = p * π (a^3/3) / ∞ is undefined, we see that there is no direct relationship between the volume density p and the line density λ for an infinitely long solid cylinder.