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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✍️Mathematical Proof
$\complement\cdots$Counselor
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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.
✍️Mathematical Proof
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🧄
- Proving the Cross Product Rules with the Levi-Civita Symbol
- Proving the Epsilon-Delta Relation and the Bac-Cab Rule
- Simplifying Levi-Civita and Kronecker Delta Identities
- Dot Cross and Triple Products
- Why a Cube's Diagonal Angle Never Changes
- How the Cross Product Relates to the Sine of an Angle
- Finding the Shortest Distance and Proving Orthogonality for Skew Lines
- A Study of Helical Trajectories and Vector Dynamics
- The Power of Cross Products: A Visual Guide to Precessing Vectors
- Divergence and Curl Analysis of Vector Fields
- Unpacking Vector Identities: How to Apply Divergence and Curl Rules
- Commutativity and Anti-symmetry in Vector Calculus Identities
- Double Curl Identity Proof using the epsilon-delta Relation
- The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation
- Surface Parametrisation and the Verification of the Gradient-Normal Relationship
- Proof and Implications of a Vector Operator Identity
- Conditions for a Scalar Field Identity
- Solution and Proof for a Vector Identity and Divergence Problem
- Kinematics and Vector Calculus of a Rotating Rigid Body
- Work Done by a Non-Conservative Force and Conservative Force
- The Lorentz Force and the Principle of Zero Work Done by a Magnetic Field
- Calculating the Area of a Half-Sphere Using Cylindrical Coordinates
- Divergence Theorem Analysis of a Vector Field with Power-Law Components
- Total Mass in a Cube vs. a Sphere
- Momentum of a Divergence-Free Fluid in a Cubic Domain
- Total Mass Flux Through Cylindrical Surfaces
- Analysis of Forces and Torques on a Current Loop in a Uniform Magnetic Field
- Computing the Integral of a Static Electromagnetic Field
- Surface Integral to Volume Integral Conversion Using the Divergence Theorem
- Circulation Integral vs. Surface Integral
- Using Stokes' Theorem with a Constant Scalar Field
- Verification of the Divergence Theorem for a Rotating Fluid Flow
- Integral of a Curl-Free Vector Field
- Boundary-Driven Cancellation in Vector Field Integrals
- The Vanishing Curl Integral
- Proving the Generalized Curl Theorem
- Computing the Magnetic Field and its Curl from a Dipole Vector Potential
- Proving Contravariant Vector Components Using the Dual Basis
- Verification of Orthogonal Tangent Vector Bases in Cylindrical and Spherical Coordinates
- Vector Field Analysis in Cylindrical Coordinates
- Vector Field Singularities and Stokes' Theorem
- Compute Parabolic coordinates-related properties
- Analyze Flux and Laplacian of The Yukawa Potential
- Verification of Vector Calculus Identities in Different Coordinate Systems
- Analysis of a Divergence-Free Vector Field
- The Uniqueness Theorem for Vector Fields
- Analysis of Electric Dipole Force Field
🧄Proof and Derivation-2
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