Physics · Chapter 23
Study notes aligned to the official NEB syllabus.
A direct current (DC) circuit is a closed conducting path in which charge flows steadily in one direction, driven by a source of constant potential difference such as a cell or a battery. This chapter builds up the physics of such circuits: what electric current is at the level of moving charge carriers, how current relates to the applied potential difference through Ohm's law, how the material and dimensions of a conductor fix its resistance, how resistors combine in series and parallel, how a potential divider produces a variable output voltage, what electromotive force and internal resistance mean, and finally how energy and power are delivered and dissipated in a circuit.
The core relations used throughout are:
$$ \begin{aligned} I &= \frac{Q}{t} \ \qquad V &= IR \ \qquad P &= VI \end{aligned} $$
When a potential difference is applied between two points of a conductor, an electric field is set up inside it. This field exerts a force on the free charge carriers (electrons in a metal), making them drift slowly through the conductor. This ordered flow of charge is the electric current.
Electric current is defined as the rate of flow of electric charge through a cross-section of a conductor:
$$I = \frac{Q}{t}$$
where $Q$ is the charge (in coulombs) that crosses the section in time $t$ (in seconds). By convention the direction of current is the direction of flow of positive charge, which is opposite to the direction in which the electrons actually move. Current is a scalar quantity. Its SI unit is the ampere (A), equal to one coulomb per second ($1\ \text{A} = 1\ \text{C s}^{-1}$), and it is measured with an ammeter connected in series.
One ampere: the current through a conductor is one ampere when one coulomb of charge flows through any cross-section of it in one second.
Drift velocity ($v_d$) is the average velocity with which the free electrons of a conductor move under the action of an applied electric field. Although individual electrons move rapidly and randomly (of the order of $10^5\ \text{m s}^{-1}$ due to thermal motion), their net drift superimposed by the field is very small, typically of the order of $10^{-4}\ \text{m s}^{-1}$.
Consider a conductor of cross-sectional area $A$ and length $l$. Let $n$ be the number of free charge carriers per unit volume (the number density), and let each carrier have charge $q$ (for electrons, $q = e = 1.6 \times 10^{-19}\ \text{C}$).
Volume of the conductor:
$$V = A l$$
Total number of free carriers in it:
$$ \begin{aligned} N &= n \times V \ &= n A l \end{aligned} $$
Total free charge available to move:
$$Q = N q = n A l, q$$
When the field is applied, suppose the carriers drift with velocity $v_d$, so that all the carriers in the length $l$ cross the far end in time:
$$t = \frac{l}{v_d}$$
The current is the total charge divided by this time:
$$ \begin{aligned} I &= \frac{Q}{t} \ &= \frac{n A l, q}{,l / v_d,} \ \boxed{I = n A v_d, q} \end{aligned} $$
For electrons this is written $I = nAv_d e$. Rearranging gives the drift velocity in terms of the current:
$$v_d = \frac{I}{n A q}$$
The current density $J$ is the current flowing per unit cross-sectional area, taken normal to the flow:
$$J = \frac{I}{A}$$
Using $I = nAv_dq$ gives $J = n v_d q$. Current density is a vector directed along the flow of positive charge; its SI unit is $\text{A m}^{-2}$. The current itself, obtained from $I = \vec{J}\cdot\vec{A}$ (a scalar product), is a scalar quantity.
Problem: An electron moves in a circle of radius $10\ \text{cm}$ with a constant speed of $5 \times 10^{6}\ \text{m s}^{-1}$. Find the electric current at a point on the circle.
Given:
Formula: The electron passes a given point once per revolution, so it constitutes a current equal to the charge times the frequency of revolution $f = \dfrac{v}{2\pi r}$:
$$ \begin{aligned} I &= e f \ &= \frac{e v}{2\pi r} \end{aligned} $$
Substitution:
$$ \begin{aligned} I &= \frac{(1.6 \times 10^{-19})(5 \times 10^{6})}{2 \times 3.14 \times 0.1} \ &= \frac{8 \times 10^{-13}}{0.628} \ I &= 1.27 \times 10^{-12}\ \text{A} \end{aligned} $$
Answer: The electric current at a point on the circle is approximately $\mathbf{1.27 \times 10^{-12}\ A}$.
Ohm's law states that the current flowing through a conductor is directly proportional to the potential difference applied across its ends, provided the physical conditions (especially temperature) remain constant:
$$I \propto V \quad\Longrightarrow\quad V = IR$$
The constant of proportionality $R$ is the resistance of the conductor.
Electrical resistance is the property of a conductor by which it opposes the flow of charge through it. From Ohm's law it is the ratio of the potential difference across the conductor to the current through it:
$$R = \frac{V}{I}$$
Its SI unit is the ohm ($\Omega$), where $1\ \Omega = 1\ \text{V A}^{-1}$.
One ohm: a conductor has a resistance of one ohm if a current of one ampere flows through it when a potential difference of one volt is applied across its ends.
A direct current (DC) circuit is a closed conducting path in which charge flows steadily in one direction, driven by a source of constant potential difference such as a cell or a battery. This chapter builds up the physics of such circuits: what electric current is at the level of moving charge carriers, how current relates to the applied potential difference through Ohm's law, how the material and dimensions of a conductor fix its resistance, how resistors combine in series and parallel, how a potential divider produces a variable output voltage, what electromotive force and internal resistance mean, and finally how energy and power are delivered and dissipated in a circuit.
The core relations used throughout are:
When a potential difference is applied between two points of a conductor, an electric field is set up inside it. This field exerts a force on the free charge carriers (electrons in a metal), making them drift slowly through the conductor. This ordered flow of charge is the electric current.
Electric current is defined as the rate of flow of electric charge through a cross-section of a conductor:
where is the charge (in coulombs) that crosses the section in time (in seconds). By convention the direction of current is the direction of flow of positive charge, which is opposite to the direction in which the electrons actually move. Current is a scalar quantity. Its SI unit is the ampere (A), equal to one coulomb per second (), and it is measured with an ammeter connected in series.
One ampere: the current through a conductor is one ampere when one coulomb of charge flows through any cross-section of it in one second.
Drift velocity () is the average velocity with which the free electrons of a conductor move under the action of an applied electric field. Although individual electrons move rapidly and randomly (of the order of due to thermal motion), their net drift superimposed by the field is very small, typically of the order of .
Consider a conductor of cross-sectional area and length . Let be the number of free charge carriers per unit volume (the number density), and let each carrier have charge (for electrons, ).
Volume of the conductor:
Total number of free carriers in it:
Total free charge available to move:
When the field is applied, suppose the carriers drift with velocity , so that all the carriers in the length cross the far end in time:
The current is the total charge divided by this time:
For electrons this is written . Rearranging gives the drift velocity in terms of the current:
The current density is the current flowing per unit cross-sectional area, taken normal to the flow:
Using gives . Current density is a vector directed along the flow of positive charge; its SI unit is . The current itself, obtained from (a scalar product), is a scalar quantity.
Problem: An electron moves in a circle of radius with a constant speed of . Find the electric current at a point on the circle.
Given:
Formula: The electron passes a given point once per revolution, so it constitutes a current equal to the charge times the frequency of revolution :
Substitution:
Answer: The electric current at a point on the circle is approximately .
Ohm's law states that the current flowing through a conductor is directly proportional to the potential difference applied across its ends, provided the physical conditions (especially temperature) remain constant:
The constant of proportionality is the resistance of the conductor.
Electrical resistance is the property of a conductor by which it opposes the flow of charge through it. From Ohm's law it is the ratio of the potential difference across the conductor to the current through it:
Its SI unit is the ohm (), where .
One ohm: a conductor has a resistance of one ohm if a current of one ampere flows through it when a potential difference of one volt is applied across its ends.