Themes > Science > Physics > Electromagnetism > Electrostatics > Batteries, current, and Ohm's law > Electric Current; Resistance

We have seen that a positive charge will move from a higher to a lower electrostatic potential. Subjecting charges to appropriate potential configurations gives us the means to control their movement. The movement of electric charge is called an electric current, and is denoted by $I$. The electrostatic potential is the cause, and the electric current is the effect. Why are we interested in electrical currents? One of the over-riding considerations in our study of nature is to find new forms of energy, such as the electrostatic potential energy. The next question that naturally arises is whether we can transform electrical energy into other forms of energy? In particular, can we move objects and power our homes and industries with this (new) form of energy? Recall work adds to energy, and is one way of transforming energy from one form to another. To do work, we must apply a force $F$ on an object over some distance $d$, with the resulting work done given by $Fd$. Movement in space under the action of the electrical force is hence essential in converting electrical energy into mechanical energy. Only electrical charges can be moved by electrical forces, or what is the same, by electric fields. An electrical current is an instance of such a movement of electric charges, and is therefore a necessary step in doing electrical work. Consider a stream of charges flowing past a fixed point. Consider an imaginary surface $S$ placed orthogonal to the flow of charges. The flow of charge is measured by the current $I$, which is the amount of charge that flows across the surface $S$ per unit area and in unit time interval . Current is given by the charge flux $\Delta q =q_2-q_1$ in time $\Delta t=t_2-t_1$. Hence
\begin{displaymath} I=\frac{q_2-q_1}{t_2-t_1}\equiv \frac{\Delta q}{\Delta t} \end{displaymath} (7.22)

The dimensions of current is $[I]=CT^{-1}$, and the SI unit of current is Amperes, denoted by $A$. That is
\begin{displaymath} 1 \mathrm{ampere}= 1 \mbox{\rm {coulomb per second}} \end{displaymath} (7.23)

Electric current $I$ in general depends on the surface through which the flow of charge is taking place. In one-dimension $I$ has a sign, with say $+I$ signifying current flowing in one direction and $-I$ for flow in the opposite one. Conservation of charge holds good for moving charges, and is expressed by the laws of current conservation. If a current $I$ flowing in a wire bifurcates into two wires with currents, say $I_1$ and $I_2$, then charge conservation requires
\begin{displaymath} I=I_1+I_2 \end{displaymath} (7.24)

So how does one go about creating the flow of an electrical current? Since we first need to get our hands on some electrons, we start with a conductor, say a copper wire, since it has a supply of free electrons. We next need to create an electric field that will exert a force on electric charges and make them move. To create a constant electric field, all we need to do is to create an electrostatic potential difference between two points in the wire; this will in turn exert force on the charges so as to create a current. If the potential difference between two points in a wire separated by distance $d$ is $V$, the electric field along the wire is then given by
\begin{displaymath} E=\frac{V}{d} \end{displaymath} (7.25)

An electrical battery is a device that creates such an electrostatic potential difference, called the voltage $V$, between its two terminals. The battery converts chemical energy into electrical energy. Figure 7.11 shows a copper wire connected in a closed circuit with a resistance $R$ and a battery with voltage $V$. The electrostatic potential, say $V_+$ at the positive terminal(+) is taken to be higher than the electrostatic potential, say $V_-$ at the negative terminals (-); the voltage of the battery is the potential difference given by $V=V_+-V_-$. Due to the potential difference $V$, free charges in the conductor (copper wire) are set in motion and yield a current $I$ in the circuit. The convention in labeling the terminals of a battery means that the positive charges flows from the higher potential at the positive terminal to the lower potential at the negative terminal. Consider piped water that is pumped to a height $h$, and flows back to the ground. The electrical current $I$ can be compared to the kinetic energy of the water as it hits the ground. The battery is the analogue of the pump, the potential difference $V$ created by the battery is the analogue of the height $h$ to which the water is raised. The resistance $R$ is analogous to the radius of the pipe, and whether it is clean, or whether there is accumulation of gravel and sand in the pipe that hinders, or even blocks, the flow of water.

 

Figure 7.11: A Circuit with Resistance and Voltage
\begin{figure} \begin{center} \epsfig{file=core/circuit3.eps, width=4cm} \end{center} \end{figure}

 

What is the relation between the current $I$ in the circuit, the resistance $R$ and the voltage $V$? From the analogy of water flow, we expect that the larger the voltage $V$ the higher the current, and the higher the value of the resistance $R$ the lower the current. We hence expect that
\begin{displaymath} I=\frac{V}{R} \mbox{\rm { : Ohm's Law}} \end{displaymath} (7.26)

Ohm's law states that if we create a potential difference of $V$ in a conductor having resistance $R$, then a current $I$ given by eq.(7.33) will flow in the conductor. The dimension of resistance is $[R]=ML^2T^{-1}C^{-2}$; the SI unit of resistance is ohms and
$\displaystyle 1\mathrm{ohm}$ $\textstyle =$ $\displaystyle 1\mbox{\rm { volt per ampere}}$ (7.27)
  $\textstyle =$ $\displaystyle 1V/A$ (7.28)

Electrical currents can flow in vacuum as well as in a medium. The resistance $R$ measures the relative ease with which a currents flows in a medium. Let a material of length $L$ and area $A$ have resistance of $R$; the resistivity $\rho$ of the material is defined by

\begin{displaymath} R=\rho\frac{L}{A} \end{displaymath} (7.29)

Note $\rho$ has units of ohm-m. Table 7.1 shows the resistivity of different kinds of material. Resistance also depends on temperature, and increases with increasing temperature.
Table 7.1: Resistivity at Temperature $20^0$C
Material $\rho$
  ohm-meter
Conductors  
Silver $1.59\times 10^{-8}$
Copper $1.68\times 10^{-8}$
Mercury $98\times 10^{-8}$
Semiconductors  
Graphite (carbon) $(3-60)\times 10^{-5}$
Germanium $(1-500)\times 10^{-3}$
Silicon $0.1-60$
Insulators  
Glass $10^9-10^{12}$
Rubber $10^{13}-10^{15}$


Worthy to note is that superconductors are materials whose properties are explicable only by quantum theory, and have zero resistance to the flow of electrical currents. Conductors and insulators allow and block the flow of currents respectively, and can be understood based on principles of classical physics. Semiconductors lie in-between conductors and insulators, and their workings are based on quantum theory. Conductors have electrons that are not bound to the nuclei, and can hence move in response to an external electric field or a potential difference. All parts of a conductor at equilibrium are at the same potential, since any potential difference would cause currents to flow. One can have many resistors, and which can be combined in series and in parallel, with the value of the resultant resistance being derivable from Ohm's law. Energy is constantly lost due to the heating of the resistance $R$, called Ohmic heating. How much power is expended by the battery in keeping the current flowing in the circuit? In $\Delta t$ time, charge of amount $\Delta q$ flows from the high electrostatic potential to the lower potential, hence losing potential energy $\Delta U$. The difference in the electrostatic potential is given by the voltage of the battery, namely $V$, and hence
\begin{displaymath} \Delta U=\Delta q V \end{displaymath} (7.30)

Power is rate of loss of energy, in this case of potential energy $U$; hence
$\displaystyle P$ $\textstyle =$ $\displaystyle \frac{\Delta U}{\Delta t}$ (7.31)
  $\textstyle =$ $\displaystyle \frac{\Delta q}{\Delta t}V$ (7.32)
$\displaystyle \Rightarrow P$ $\textstyle =$ $\displaystyle IV$ (7.33)
  $\textstyle =$ $\displaystyle I^2R=\frac{V^2}{R}$ (7.34)

Note that the power loss in a circuit is proportional to $I^2$. In other words, if we reverse the flow of current, the sign of $I$ will change to $-I$, but the loss due to Ohmic heating will be the same, no matter in which direction the current flows. The loss of power of the battery due to heating is not a net loss, but rather an example of the transformation of energy. Light bulbs that use a filament emanate light when the filament is heated by the flow of current, and over 95% of the power $I^2R$ is converted into light, and similarly for electric ovens and so on. Hence, electrical currents serve as a vehicle for converting the energy expended by the battery into other forms of energy. The current we have considered so far is called a D.C. (Direct Current) since its direction does not change. An A.C.(Alternating Current) is one in which the current changes direction, and hence its sign, with some fixed frequency.


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