Divergence and Curl of Electrostatic Fields

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20 r

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Field of a point charge

arrows field lines:E lines/area ofnumber

of direction E

Bits of thread in oil align with the field lines.

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Construction principle:Field lines start at +q and end at –q.The number of lines is proportional to q.If the total charge is different from 0, there will be linesgoing to or coming from infinity.Field lines never cross.The field lines point in radial direction near point charges.

Divergence of E (Gauss’s Law)

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Application of Gauss’s law (integral form):

1. Spherical symmetry:

3. Plane symmetry:

The direction of the field is known, and it is constant on the Gaussian surface

2. Cylindrical symmetry:

Example 2.2What is the field outside the uniformly charged sphere?

Example 2.3Field inside a long cylinder with charge density ks

Example 2.4Infinite plate with uniform surface charge density

Example 2.5

Field Lines and Flux

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aE dE

Flux through a surface:

Streamlines of a fluid flowing around a cylinder.

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Conservation of flux:

00 aEE d

Flux tube:

The flux (number of field/stream lines) through the different cross sections of the flux tube is constant.

Curl of E

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r

q

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General properties

Laws of electrostatics

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0 E

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VS

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enc , :form integral aE