Using the divergence theorem, the l h s of the equation is written as a volume integration /sJda=V·Jd7, so the conservation equation is V·Jar h pd Since this is valid for an T is arbitrary V·J This is the conservation of charge stated in differential orm
Homogeneous Conductor in Steady state If the system is in steady state, -dtp=0, thus V·J=0 Then by j=oE, one has V·E=0 If the conductor is also homogeneous, o is independent of position one has V·E=0. Thus by v.E=p/∈0, i.e. under steady-state condition the net charge den sity inside a homogeneous conductor carrying a current s zero
5.2 Ohm's Law Given an element of conductor of length l and cross section a If a potential difference v is maintained btwn its two ends there appears a current I From -oE withJ=I/A andE=v/1, one has A so the current is given by the following Where the resistance is 1≡ A --Ohm's Law
Remarks (1)The current thru a resistor is a the voltage across 止t (2 )A resistor satisfying this law is said to be linear (3)The current and voltage are measured as in Fig.5-4 The current thru a voltmeter is usually assumed to be negligible compares with that thru R R Figure 5.4 Measurement of the current I through a resistor R, and of the voltage v across it. A circle marked I represents an ammeter, and a circle marked v,a voltmeter