329lect16 - 16 Charge conservation continuity eqn...

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16 Charge conservation, continuity eqn, displace- ment current, Maxwell’s equations Total electric charge is conserved in nature in the following sense: if a process generates (or eliminates) a positive charge, it always does so as accompanied by a negative charge of equal magnitude. Example: Photoionization of atoms and molecules can generate free positive ions and free negative electrons in pairs (see margin figure). Photoionization is a process that converts bound charge carriers into free charge carriers. Example: Recombination when a positive ion and an electron get together to produce a charge neutral atom or molecule. Example: Annihilation of an electron (negative charge) by a positron (positive charge of equal magnitude). Q V = 0 V S t = 0 (a) At t=0 volume V contains a neutral atom but no net charge Q V = 0 V S t = t 1 (b) At t=t1 volume V contains a proton and a free electron after the ionization of the hydrogen atom. There is still no net charge in the volume. H -e e Q V = e V S t = t 2 (c) At t=t2 volume V now contains only a proton after the exit of free electron through surface S. Now V contains a net charge e. -e e As a consequence, if the total electric charge Q V contained in any finite volume V changes as a function of time, this change must be attributed to a net transport of charge, i.e., electric current, across the bounding surface S of volume V as detailed below. 1
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Consider two distinct surfaces S 1 and S 2 bounded by the same closed loop C (as shown in the margin) such that a volume V is contained between the two surfaces. Let I 1 = S 1 J · d S 1 and I 2 = S 2 J · d S 2 denote currents flowing through surfaces S 1 and S 2 , respectively. V S 1 S 2 C d S 1 d S 2 d S Note that current I 1 through surface S 1 enters volume V , while current I 2 through surface S 2 exits volume V (with the directions assigned to d S 1 and d S 2 ).
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