To determine the surface charge density of a disc taken from a uniformly charged cylinder, we first need to understand the relationship between the total charge, the volume of the cylinder, and how that charge is distributed across its cross-sectional area. Let's break this down step by step.
Understanding the Cylinder's Charge Distribution
Consider a cylinder with a uniform volume charge density, denoted as ρ (rho). The total charge (Q) within the cylinder can be expressed as:
Q = ρ × V
Here, V is the volume of the cylinder, which can be calculated using the formula for the volume of a cylinder:
V = πR²L
Substituting this into the equation for total charge gives:
Q = ρ × πR²L
Analyzing the Disc
Now, let’s focus on a thin disc of thickness dx taken from this cylinder. The volume of this disc can be calculated as:
V_disc = πR²dx
Since the charge is uniformly distributed, the charge (dQ) contained in this thin disc can be expressed as:
dQ = ρ × V_disc = ρ × πR²dx
Calculating the Surface Charge Density
The surface charge density (σ) of the disc is defined as the charge per unit area. The area (A) of the circular face of the disc is:
A = πR²
σ = dQ / A
Substituting the expressions for dQ and A into this equation gives:
σ = (ρ × πR²dx) / (πR²) = ρdx
Identifying the Correct Option
Now, looking at the options provided:
- (1) rdx
- (2) rLdx/R
- (3) rRdx/L
- (4) rπR²/L
Here, "r" represents the volume charge density ρ. From our calculations, we see that the surface charge density of the disc is indeed:
σ = ρdx
Thus, the correct answer is option (1) rdx.
Conclusion
In summary, by analyzing the charge distribution in a uniformly charged cylinder and focusing on a thin disc, we derived the surface charge density effectively. This approach not only clarifies the concept but also reinforces the importance of understanding how charge is distributed in three-dimensional objects.
