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

\[ \begin{aligned} &V \propto I\\ &\text {or, } V=R I \end{aligned} \] (3.3)

where the constant of proportionality \( R \) is called the \( resistance \) of the conductor. The SI units of resistance is \( ohm \), and is denoted by the symbol \( \Omega \). 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 Eq. (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 \( R_C \) is [from Eq. (3.3)],

\[ R_C=\frac{2V}{I}=2R \](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,

\[ R \propto l \](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)].

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 \( R_1 \) is

\[ R_{1}=\frac{V}{(I / 2)}=2 \frac{V}{I}=2 R \](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,

\[ R \propto \frac{1}{A} \](3.7)

Combining Eqs. (3.5) and (3.7), we have

\[ R \propto \frac{l}{A} \](3.8)

and hence for a given conductor

\[ R =\rho \frac{l}{A} \](3.9)

where the constant of proportionality \( \rho \) depends on the material of the conductor but not on its dimensions. \( \rho \) is called resistivity.

Using the last equation, Ohm’s law reads

\[ V=I \times R=\frac{I \rho l}{A} \](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 \( El \). Using these, the last equation reads

\[ \begin{array}{l} \mathrm{E} \boldsymbol{l}=\boldsymbol{j} \rho \boldsymbol{l} \\ \text{or } \boldsymbol{E}=\boldsymbol{j} \rho \end{array} \](3.11)

The above relation for magnitudes E and \( j \) can indeed be cast 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 \( E \), and is also a vector \( \mathbf{j}(\equiv j \mathbf{E} / \mathrm{E}) \). Thus, the last equation can be written as,

\[ \mathbf{E}=\mathbf{j} \rho \](3.12)

\[ \mathbf{j}=\sigma \mathbf{E} \](3.13)

where \( \sigma \equiv 1 / \rho \) is called the conductivity. Ohm’s law is often stated in an equivalent form, Eq. (3.13) in addition to Eq.(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.