The Yukawa Potential: Analysis of Flux and Laplacian

The divergence theorem must be carefully adapted to handle singularities within a volume. While a direct surface integral readily provides the correct physical flux, the divergence theorem's volume integral needs to account for the singularity (like a point charge at the origin) using a delta function to yield a consistent result. This highlights that the Laplacian of a potential is directly linked to the charge density, with the delta function representing a point charge.

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Direct Surface Integral vs. Divergence Theorem

The key takeaway is that applying the divergence theorem for a field with a singularity inside the volume is not straightforward and requires careful consideration of that singularity's contribution.

The flux calculated by the direct surface integral, $\Phi=q e^{-k R}(1+k R)$, is the correct physical result.

The Role of the Singularity

The discrepancy between the two methods initially arose because the divergence theorem in its basic form doesn't account for a point singularity at the origin. The Laplacian of the Yukawa potential, $\nabla^2 \phi=\frac{k^2 q e^{-k r}}{4 \pi r}$, is infinite at $r=0$. Physically, this infinite value represents a point charge at the origin.

To correctly use the divergence theorem, the singular behavior must be included mathematically as a delta function, which represents the source charge. Once this is accounted for, the volume integral of the divergence yields the same result as the surface integral, validating both methods.

Laplacian of the Potential

The Laplacian of the Yukawa potential, $\nabla^2 \phi$, is directly related to the charge density via Poisson's equation. The analysis shows that for a screened potential, the Laplacian is not zero everywhere. At the origin, it contains a delta function, indicating the presence of a point charge, while for $r>0$, it is a function of $r$, representing a distributed, "screened" charge density.

🎬Demonstration

The visualization illustrates how it visually demonstrates the concept of a screened force. Unlike a long-range force, like electromagnetism, a screened force has a finite range and its influence diminishes rapidly with distance.

how different parameters affect Yukawa Potential and its Vector Field