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Which of the following identities is always zero for static fields?
Grad(Curl V)
Curl(Div V)
Div(Grad V)
Curl(Grad V)
Curl(Grad V)
The identity ∇×(∇V)=0 is a fundamental vector calculus identity that holds true for any scalar field V that is twice continuously differentiable. In electromagnetics, this implies that the curl of a static electric field (the gradient of an electric scalar potential) is always zero, consistent with the conservative nature of electrostatic fields.
The identity ∇×(∇V)=0 is a fundamental vector calculus identity that holds true for any scalar field V that is twice continuously differentiable. In electromagnetics, this implies that the curl of a static electric field (the gradient of an electric scalar potential) is always zero, consistent with the conservative nature of electrostatic fields.
E=−∇V — Relation between static electric field and potential
∇×E=0 — Conservative property of static electric fields
The principle arises from the fact that the partial derivatives of a smooth scalar function commute (fracpartial2Vpartialxpartialy=fracpartial2Vpartialypartialx). When applying the curl operator to the gradient, the mixed partial derivatives subtract to zero identically.
The gradient of any scalar field is an irrotational vector field.
A conservative field is one where the work done moving a charge along a closed path is zero.
This identity is central to defining the electrostatic potential V.
Allows for scalar potential representation of vector fields
Simplifies Maxwell's equations for static conditions
Does not apply to time-varying fields where ∇×E=−∂t∂B
Electrostatics (calculation of Potential)
Fluid Dynamics (irrotational flow analysis)
Option A: ∇(∇×A)=0 is the divergence of a curl, which is always zero due to the symmetry of the cross product operator.
Option C: ∇⋅(∇V)=∇2V (Laplacian of V), which is not necessarily zero (e.g., in a region with charge density).
D is correct — The curl of the gradient of any scalar field is mathematically equivalent to the zero vector.
Remember that in time-varying fields (Electrodynamics), the curl of the electric field is non-zero due to Faraday's Law; always check if the problem specifies 'static' conditions.