Using the divergence theorem the l.h. s of the equation is written as a volume integration /sJda=V·Jdr, so the conservation equation is 1V·Jr=-mFpr. Since this is valid for an t is arbitrary This is the conservation of charge stated in differential orm
Homogeneous Conductor in Steady state If the system is in steady state, d=0, thus V·J=0. Then by j =oE, one has V·aE=0. If the conductor is also homogeneous o is independent ot 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 with=I/A andE=v/l, one has a l so the current is given by the following where the resistance is A -Ohm's law
Remarks (1)The current thru a resistor is a the voltage across (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 across it. a circle marked I represents an ammeter, and a circle marked V,a voltmeter