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www.GOALias.blogspot.comPhysicsGEORG SIMON OHM (1787–1854)96Georg Simon Ohm (1787–1854) German physicist,professor at Munich. Ohmwas led to his law by ananalogy between theconduction of heat: theelectric field is analogous tothe temperature gradient,and the electric current isanalogous to the heat flow.identical to the first and the same current I flows throughboth. The potential difference across the ends of thecombination is clearly sum of the potential differenceacross the two individual slabs and hence equals 2V. Thecurrent through the combination is I and the resistanceof the combination R Cis [from Eq. (3.3)],2VRC= = 2 R(3.4)Isince V/I = R, the resistance of either of the slabs. Thus,doubling the length of a conductor doubles theresistance. In general, then resistance is proportional tolength,R∝ l(3.5)Next, imagine dividing the slab into two by cutting itlengthwise so that the slab can be considered as acombination of two identical slabs of length l, but eachhaving a cross sectional area of A/2 [Fig. 3.2(c)].For a given voltage V across the slab, if I is the currentthrough the entire slab, then clearly the current flowingthrough each of the two half-slabs is I/2. Since thepotential difference across the ends of the half-slabs is V,i.e., the same as across the full slab, the resistance of eachof the half-slabs R 1isV VR = 1 2 2 R.( I/2)= I= (3.6)Thus, halving the area of the cross-section of a conductor doublesthe resistance. In general, then the resistance R is inversely proportionalto the cross-sectional area,1R ∝ (3.7)ACombining Eqs. (3.5) and (3.7), we havelR ∝ (3.8)Aand hence for a given conductorlR = ρ(3.9)Awhere the constant of proportionality ρ depends on the material of theconductor but not on its dimensions. ρ is called resistivity.Using the last equation, Ohm’s law readsIρlV = I × R= (3.10)ACurrent per unit area (taken normal to the current), I/A, is calledcurrent density and is denoted by j. The SI units of the current densityare A/m 2 . Further, if E is the magnitude of uniform electric field in theconductor whose length is l, then the potential difference V across itsends is El. Using these, the last equation reads