# 4.2: Resistance and Ohm's Law

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- 5432

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)When a potential difference is maintained across the electrodes in an electrolytic cell, a current flows through the electrolyte. This current is carried by positive ions moving from the positive electrode towards the negative electrode and also, simultaneously, by negative ions moving from the negative electrode towards the positive electrode. The conventional direction of the flow of electricity is the direction in which positive charges are moving. That is to say, electricity flows from the positive electrode towards the negative electrode. The positive ions, then, are moving in the same direction as the conventional direction of flow of electricity, and the negative ions are moving in the opposite direction.

When current flows in a *metal*, the current is carried exclusively by means of negatively charged electrons, and therefore the current is carried exclusively by means of particles that are moving in the opposite direction to the conventional flow of electricity. Thus “electricity” flows from a point of high potential to a point of lower potential; electrons move from a point of low potential to a point of higher potential.

When a potential difference \(V\)is applied across a resistor, the ratio of the potential difference across the resistor to the current \(I\) that flows through it is called the *resistance*, \(R\), of the resistor. Thus

\[V = IR.\label{4.2.1}\]

This equation, which defines resistance, appears at first glance to say that *the current through a resistor is proportional to the potential difference across it*, and this is *Ohm’s Law.* Equation \ref{4.2.1}, however, implies a simple proportionality between \(V\) and \(I\)only if \(R\) is constant and independent of \(I\) or of \(V\). In practice, when a current flows through a resistor, the resistor becomes hot, and its resistance increases – and then \(V\) and \(I\) are no longer linearly proportional to one another. Thus one would have to state Ohm’s Law in the form that *the current through a resistor is proportional to the potential difference across it, provided that the temperature is held constant*. Even so, there are some substances (and various electronic devices) in which the resistance is not independent of the applied potential difference even at constant temperature. Thus it is better to regard Equation \ref{4.2.1} as a definition of resistance rather than as a fundamental law, while also accepting that it is a good description of the behaviour of most real substances under a wide variety of conditions as long as the temperature is held constant.

Definition: resistance and Conductance

If a current of one amp flows through a resistor when there is a potential difference of one volt across it, the resistance is one *ohm* (\(\Omega\)). (Clear though this definition may appear, however, recall from chapter 1 that we have not yet defined exactly what we mean by an amp, nor a volt, so suddenly the meaning of “ohm” becomes a good deal less clear! I do promise a definition of “amp” in a later chapter – but in the meantime I crave your patience.)

The *dimensions* of resistance are

\[\dfrac{\text{ML}^2\text{T}^{-2}\text{Q}^{-1}}{\text{T}^{-1}\text{Q}} = \text{ML}^2\text{T}^{-1}\text{Q}^{-2}.\]

The reciprocal of resistance is *conductance*, \(G\)*.* Thus \(I = GV\). It is common informal practice to express conductance in “mhos”, a “mho” being an ohm^{-1}. The official SI unit of conductance, however, is the siemens (S), which is the same thing as a “mho”, namely one A V^{-1}.

The *resistance* of a *resistor* is proportional to its length \(l\) and inversely proportional to its cross-sectional area *A*:

\[R = \dfrac{\rho l}{A} \label{4.2.2}\]

The constant of proportionality \(\rho\) is called the *resistivity* of the material of which the resistor is made. Its dimensions are ML^{3}T^{-1}Q^{-2}, and its SI unit is ohm metre, or \(\Omega\) m.

The reciprocal of resistivity is the *conductivity*, \(\sigma\). Its dimensions are M^{-1}L^{-3}TQ^{2}, and its SI unit is siemens per metre, S m^{-1}.

For those who enjoy collecting obscure units, there is an amusing unit I once came across, namely the unit of surface resistivity. One is concerned with the resistance of a thin sheet of conducting material, such as, for example, a thin metallic film deposited on glass. The resistance of some rectangular area of this is proportional to the length *l* of the rectangle and inversely proportional to its width *w*:

\[R = \dfrac{\rho l}{w}\nonumber \]

The resistance, then, depends on the ratio \(l/w\) – i.e. on the shape of the rectangle, rather than on its size. Thus the resistance of a 2 mm \(\times\) 3 mm rectangle is the same as that of a 2 m \(\times\) 3 m rectangle, but quite different from that of a 3 mm \(\times\) 2 mm rectangle. The surface resistivity is defined as the resistance of a rectangle of unit length and unit width (i.e. a square) – and it doesn’t matter what the size of the square. Thus the units of surface resistivity are ohms per square. (End of sentence!)

As far as their resistivities are concerned, it is found that substances may be categorized as *metals*, *nonconductors* (insulators), and *semiconductors*. Metals have rather low resistivities, of the order of 10^{-8} \(\Omega\) m. For example:

Silver: 1.6 \(\times\) 10^{-8} \(\Omega\) m

Copper: 1.7 \(\times\) 10^{-8}

Aluminium: 2.8 \(\times\) 10^{-8}

Tungsten: 5.5 \(\times\) 10^{-8}

Iron: 10 \(\times\) 10^{-8}

Nonconductors have resistivities typically of order 10^{14} to 10^{16} \(\Omega\) m or more. That is, for most practical purposes and conditions they don’t conduct any easily measurable electricity at all.

Semiconductors have intermediate resistivities, such as

Carbon: 1500 \(\times\) 10^{-8} \(\Omega\) m

Germanium: 4.5 \(\times\) 10^{-1}

Silicon 6.4 \(\times\) 10^{+2}

There is another way, besides equation 4.2.1, that is commonly used to express Ohm’s law. Refer to Figure IV.1.

\(\text{FIGURE IV.1}\)

We have a metal rod of length \(l\), cross-sectional area *A*, electrical conductivity \(\sigma\), and so its resistance is *l*/(\(\sigma\)*A*). We clamp it between two points which have a potential difference of *V* between them, and consequently the magnitude of the electric field in the metal is \(E = V/l\). Equation 4.2.1 (Ohm’s law) therefore becomes \(El = Il/(\sigma A)\). Now introduce \(J=I/A\) as the *current density* (amps per square metre). Them Ohm’s law becomes \(J=\sigma E\). This is usually written in vector form, since current and field are both vectors, so that Ohm’s law is written

\[\textbf{J}=\sigma \textbf{E}\label{4.2.3}\]