Consider the following incorrect student statements about the divergence of the

Question

Consider the following incorrect student statements about the divergence of the electric field at point P due to a single point charge at the origin. For each statement, identify the flaw(s) in the reasoning and determine the correct conclusion using that line of reasoning. 4. "Divergence is a measure of how the magnitude of the field changes along the direction of the field, because you take the x-derivative of the x-component and so on. At point P, the electric field is growing weaker to the right, so the divergence at point P is negative." Student 1: tent 2: Divergence is a measure of the spread of the field. Since the field around point P is spreading out, the divergence at point P is positive." Student 3: "This is the electric field of a positive point charge. Gauss' Law relates the divergence of the electric field to charge, so the divergence at point P is positive."

Explanation / Answer

Student 1: Divergence of the electric field at point P is positive. As shown in the diagram the electric field has a positive spread, so to speak in the outward direction, this is shown by the directional arrows. The reason for an 'outward' direction of the elctric field is that it is a positive charge and the electric field lines of a positive charge are directed outwards from the source.

Student 2: Divergence surely is the spread of field, but spread with respect to what? With respect to a particular direction. At point P it is surely positive, but it is positive because the elctric field vector is pointing out/away at P.

Student 3: Both flux and divergence deal with flow of electric field but flux is defined over an area whereas as divergence is at a particular point. Gauss's law tells you how much volume of electric field lines is flowing out of an area whereas divergence can be compared being positive for a faucet and negative for a sink, but it only over a single point that the directional 'flow' of electic field is positive outward or inward.

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