Askiitians Tutor Team
Last Activity: 4 Months ago
To understand the behavior of the two coaxial conducting cylinders when charge densities are applied, we need to delve into some fundamental concepts of electrostatics. Let's break down each scenario you've presented: charging the inner cylinder, charging the outer cylinder, and the case of a uniform line charge.
Scenario A: Charge Density on the Inner Cylinder
When a charge density is applied to the inner cylinder, it creates an electric field in the space between the two cylinders. This electric field arises because the inner cylinder, now positively or negatively charged, will induce a charge on the inner surface of the outer cylinder. The outer cylinder will respond by redistributing its charges to maintain electrostatic equilibrium.
As a result, the outer cylinder will have an induced charge on its inner surface that is equal in magnitude but opposite in sign to that of the inner cylinder. This leads to a potential difference between the two cylinders, which can be calculated using the formula for the electric field between two cylindrical conductors:
- The electric field \( E \) in the region between the cylinders is given by \( E = \frac{\lambda}{2 \pi \epsilon_0 r} \), where \( \lambda \) is the linear charge density of the inner cylinder, \( \epsilon_0 \) is the permittivity of free space, and \( r \) is the radial distance from the axis.
- The potential difference \( V \) between the two cylinders can be found by integrating the electric field from the radius of the inner cylinder \( r_1 \) to the radius of the outer cylinder \( r_2 \):
\( V = -\int_{r_1}^{r_2} E \, dr \).
This results in a non-zero potential difference between the two cylinders when the inner cylinder is charged.
Scenario B: Charge Density on the Outer Cylinder
Now, if we apply a charge density to the outer cylinder instead, the situation changes. The outer cylinder will have a charge that induces a corresponding charge on the inner surface of the inner cylinder. However, since the inner cylinder is initially neutral, it will not have any net charge to respond with. The electric field produced by the outer cylinder will not penetrate into the inner cylinder due to its conductive nature.
In this case, the electric field inside the inner cylinder remains zero, and thus there is no potential difference between the two cylinders. The outer cylinder's charge does not affect the potential of the inner cylinder because the electric field inside a conductor in electrostatic equilibrium is always zero.
Scenario C: Uniform Line Charge
When you mention a uniform line charge \( k \), it’s important to clarify how this charge is distributed. If this line charge is placed along the axis of the inner cylinder, it will create an electric field that radiates outward. However, since the inner cylinder is conductive, it will redistribute its charges in response to this field, effectively shielding the outer cylinder from the electric field produced by the line charge.
As a result, while there may be an electric field in the space between the line charge and the inner cylinder, the potential difference between the inner and outer cylinders remains zero. The inner cylinder will adjust its surface charge to ensure that the electric field inside it is nullified, maintaining electrostatic equilibrium.
Summary of Findings
In summary:
- Charging the inner cylinder creates a potential difference between the two cylinders.
- Charging the outer cylinder results in no potential difference, as the inner cylinder remains neutral.
- A uniform line charge along the axis of the inner cylinder also does not create a potential difference due to the shielding effect of the conductive material.
Understanding these principles is crucial in electrostatics, especially when dealing with conductors and their interactions. Each scenario illustrates the fundamental behavior of electric fields and potentials in conductive materials.
