3.4 Ohm’s Law

A basic law regarding flow of currents was discovered by G.S. Ohm in 1828, long before the physical mechanism responsible for flow of currents was discovered. Imagine a conductor through which a current I is flowing and let V be the potential difference between the ends of the conductor. Then Ohm’s law states that

V ∝ I

or, V = R I .. .. (3.3)

where R, the constant of proportionality, is called the resistance of the conductor. The SI units of resistance is OHM, and is denoted by the symbol Ω. The value of the resistance R not only depends on the material of the conductor but also on the dimensions of the conductor. The dependence of R on the dimensions of the conductor can easily be determined as follows.

Consider a conductor satisfying Equation (3.3) to be in the form of a slab of length l and cross sectional area A [Fig. 3.2(a)]. Imagine placing two such identical slabs side by side [Fig. 3.2(b)], so that the length of the combination is 2l. The current flowing through the combination is the same as that flowing through either of the slabs. If V is the potential difference across the ends of the first slab, then V is also the potential difference across the ends of the second slab since the second slab is identical to the first and the same current I flows through both. The potential difference across the ends of the combination is clearly sum of the potential difference across the two individual slabs and hence equals 2V. The current through the combination is I and the resistance of the combination RC is [from Equation (3.3)],

RC = 2V/I = 2R .. .. (3.4)

since V/I = R, the resistance of either of the slabs. Thus, doubling the length of a conductor doubles the resistance. In general, then resistance is proportional to length,

R ∝ l .. .. (3.5)

Next, imagine dividing the slab into two by cutting it lengthwise so that the slab can be considered as a combination of two identical slabs of length l, but each having a cross sectional area of A/2 [Fig. 3.2(c)].

FIGURE 3.2 Illustrating the relation R = ρ l/A for a rectangular slab of length l and area of cross-section A.

For a given voltage V across the slab, if I is the current through the entire slab, then clearly the current flowing through each of the two half-slabs is I/2. Since the potential difference across the ends of the half-slabs is V, i.e., the same as across the full slab, the resistance of each of the half-slabs R1 is

R1 = V/(I/2) = 2(V/I) = 2R .. .. (3.6)

Thus, halving the area of the cross-section of a conductor doubles the resistance. In general, then the resistance R is inversely proportional to the cross-sectional area,

R ∝ 1/A .. .. (3.7)

Combining Equations (3.5) and (3.7), we have

R ∝ l/A .. .. (3.8)

and hence for a given conductor

R = ρ(l/A) .. .. (3.9)

where the constant of proportionality ρ depends on the material of the conductor but not on its dimensions. ρ is called resistivity of the material of the conductor.

Using the last equation, Ohm’s law reads

V = I × R= Iρl/A .. .. (3.10)

Current per unit area (taken normal to the current), I/A, is called current density and is denoted by j. The SI units of the current density are A/m^2. Further, if E is the magnitude of uniform electric field in the conductor whose length is l, then the potential difference V across its ends is E l, Using these, the last equation reads

E l = j ρ l

or, E = j ρ .. .. (3.11)

The above relation for magnitudes of E vector and j vector can indeed be casted in a vector form. The current density, (which we have defined as the current through unit area normal to the current) is also directed along E vector, and is also a vector j vector. Thus, the last equation can be written as,

E vector = j vector ρ .. .. (3.12)

or, j vector = σ E vector .. .. (3.13)

where σ = 1/ρ is called the conductivity of the material of the conductor. Ohm’s law is often stated in an equivalent form, Equation (3.13) in addition to Equation (3.3). In the next section, we will try to understand the origin of the Ohm’s law as arising from the characteristics of the drift of electrons.

Note carefully that current density J is a vector quantity, whereas, the current I is a scalar. T K Bansal.

START UP BLUE BOX

[Georg Simon Ohm (1787-1854) German physicist, professor at Munich. Ohm was led to his law by an analogy between the conduction of heat: the electric field is analogous to the temperature gradient, and the electric current is analogous to the heat flow.]


Accessable NCERT books by Dr T K Bansal.


END OF BLUE BOX