Two conducting thin hollow cylinders are co-aligned. The inner cylinder has a radius R1 , the outer has a radius R2 . Calculate the electric potential difference V(R2)−V(R1) between the two cylinders. The inner cylinder has a surface charge density of σa=−σ , where σ>0 , and the outer surface has a surface charge density of σb=3σ ,
The cylinders are much much longer than R1 . Thus, you may ignore end effects and neglect the thickness of the cylinders.
a. What is the electric potential difference between the outer cylinder and the inner cylinder V(R2)−V(R1)?
Express your answer in terms of R1, R2,σ, and ϵ0
b. What is the magnitude of the electric field outside the cylinders, r>R2?
Express your answer in terms of r,R1, R2,σ, and ϵ0.
c. What is the electric potential difference between a point at a distance r=2*R2 from the symmetry axis and the outer cylinder V(2*R2)−V(R2)?
Express your answer in terms of R1, R2,σ, and ϵ0.
a. Gauss law applies. I would set V=0 on the inner cylinder, since V =0 is an arbitrary location.
Then, between the cylinders, you know E due to the enclosed charge (gauss law).
Potential then is INTEGRAL of E dr. That is straightforward.
b. E outside? You know the NET charge enclosed, find E for that for an infinitely long cylinder
c. Again, now set V=0 at R2, then compute V at any r as in a.
and the aswers are?
a little more explanation please
To solve this problem, we can use the principle of superposition. We need to consider the contributions of each cylinder separately and then sum them up.
a. Electric Potential Difference between the outer and inner cylinders (V(R2) - V(R1)):
First, let's find the electric potential at a point P located at a distance r from the center of the inner cylinder (R1). Since the inner cylinder is negatively charged, it creates a potential of -σ / (2ε₀) volts at point P.
Now, let's find the electric potential at point P due to the outer cylinder. The outer cylinder is positively charged with a surface charge density of 3σ. The potential created by a uniformly charged hollow cylinder at a point outside the cylinder is V = σ / (2ε₀) * ln(R2 / R1), where ln denotes natural logarithm. At point P, the potential created by the outer cylinder is 3σ / (2ε₀) * ln(R2 / R1) volts.
The total electric potential at point P is the sum of the potentials created by the inner and outer cylinders:
V_total = (-σ / (2ε₀)) + (3σ / (2ε₀)) * ln(R2 / R1)
The electric potential difference between the outer and inner cylinders is then:
V(R2) - V(R1) = V_total - (-σ / (2ε₀))
= [(-σ / (2ε₀)) + (3σ / (2ε₀)) * ln(R2 / R1)] + (σ / (2ε₀))
= (3σ / (2ε₀)) * ln(R2 / R1)
So, the electric potential difference between the outer and inner cylinders is (3σ / (2ε₀)) * ln(R2 / R1).
b. Electric Field outside the cylinders (r > R2):
To find the electric field outside the cylinders, we can use Gauss's law. As per Gauss's law, the electric field outside a uniformly charged cylinder is given by E = σ / (ε₀ * r).
So, the magnitude of the electric field outside the cylinders (r > R2) is σ / (ε₀ * r), where r is the distance from the center of the cylinders.
c. Electric Potential Difference between a point at r = 2*R2 and the outer cylinder (V(2*R2) - V(R2)):
Using the same approach as in part a, we can find the electric potential at a point P located at a distance r = 2*R2 from the symmetry axis due to the outer cylinder. The potential created by the outer cylinder at point P can be calculated as 3σ / (2ε₀) * ln(2).
The electric potential difference between the point at r = 2*R2 and the outer cylinder is then:
V(2*R2) - V(R2) = [3σ / (2ε₀) * ln(2)] - (3σ / (2ε₀)) * ln(R2 / R1)
= [3σ / (2ε₀)] * [ln(2) - ln(R2 / R1)]
= [3σ / (2ε₀)] * ln(2R1 / R2)
So, the electric potential difference between a point at r = 2*R2 and the outer cylinder is [3σ / (2ε₀)] * ln(2R1 / R2).