saravanan13 said:Please clearly elucidate this statement "Then the x- and z-components of the curl give you z- and x-independence of Hy respectively."
The equation $\nabla\times A=0$ represents the curl of a vector field A, which is a measure of the rotation of the field. This equation specifically states that the curl of A is equal to 0, indicating that there is no rotation in the field.
A Uniform Y-Component of A means that the vector field has a constant magnitude and direction in the y-direction. This would result in a zero curl in the y-direction, as there is no change in the field's rotation in that direction. Thus, the equation $\nabla\times A=0$ would still hold true.
The physical significance of $\nabla\times A=0$ is that it represents a vector field that is irrotational, meaning there is no rotation or circular motion present in the field. This could be seen in a fluid flow, where there is no swirling motion or vortices present.
The equation $\nabla\times A=0$ is related to the concept of divergence through the fundamental theorem of vector calculus. This theorem states that for a vector field A, if the curl of A is equal to 0, then the divergence of A is also equal to 0. In other words, an irrotational field also has zero divergence.
No, the equation $\nabla\times A=0$ can only be applied to certain vector fields, specifically those that are irrotational. This means that there is no circular motion or rotation present in the field. Vector fields that do have rotation would not satisfy this equation.