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Question on Introduction to Electrodynamics divergence examples

Physics Asked on August 4, 2020

I was going through the section on divergence and became confused on these examples.
Griffith states in the textbook that (a) has positive divergence, (b) has zero divergence, and that (c) has again positive divergence.

For (c), shouldn’t the divergence be zero as all the arrows are pointing the same way? And if (c) has positive divergence, what’s the difference between (b), which also just has arrows pointing in the same direction, if not little staggered?

Thank you very much.
enter image description here

One Answer

The difference is that the vectors in (b) are all the same length / magnitude, while the vectors in (c) vary in magnitude as we move along the vertical direction.

The divergence of a vector field $vec{F}$ is $nabla cdot vec{F} = partial _x F_x + partial_y F_y + partial_z F_z$ in cartesian coordinates. If we choose the $y$ direction to be vertically upward in the figure you posted, the vector field in (c) has a $y$ component that varies as a function of the $y$ coordinate, so $partial_y F_y$ will be non-zero, and so the divergence will be non-zero.

Correct answer by d_b on August 4, 2020

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