DC-Circuits

The flow of electric charges

 DC-circuits

      The flow of charges

           The main characters

\Delta V
V
I
R

 DC-circuits

      The flow of charges

           The main characters

\Delta V
Z
X
R

Current

I
R

The underlying physics

DC-Circuits

The flow of electric charges

 DC-circuits

      The flow of charges

           Ingredients

First, you need a conductor  as a conduit for electricity

The positively charged ions in the metal are ~fixed in a lattice & the (many many) conduction-electrons bounce around like pinballs in the potential landscape.

 DC-circuits

      The flow of charges

           Ingredients

Q: What is the average speed of an electron within the metal?

A: Assuming a simple model of a 3D electron gas @ room temp: 

\frac{1}{2}m_e v^2=\frac{3}{2}k_B T
v=\sqrt{\frac{3k_B T}{m_e}}\approx 10^5 \: m/s
\text{Kinetic Energy} = \text{Thermal Energy}

 DC-circuits

      The flow of charges

           Ingredients

Q: Why are the trajectories parabolic?

A: Projectile motion

\implies \vec{a} = \frac{q_e}{m_e} \vec{E}
\implies q_e \vec{E} = m_e \vec{a}
\Delta\vec{r} = \vec{v}_0 t + \tfrac{1}{2}\vec{a}t^2

Constant acceleration in direction opposite of field,

and constant velocity perpendicular to field.

\vec{F} = m \vec{a}

Then, set up an electric field

 DC-circuits

      The flow of charges

           Ingredients

Q: What is the average "drift" velocity?

A: very slow ~

~ 10^{-4} m/s
{v}_{d} = a\ t
v_d = \frac{q_e \tau}{m_e} E

If the average time between collisions is   

 DC-circuits

      The flow of charges

           Ingredients

Then, you make a wire

 DC-circuits

      The flow of charges

           Ingredients

 DC-circuits

      The Electron Current

           Definition

\implies i_e= \frac{n_e\ A\ q_e\ \tau}{m_e} E

 DC-circuits

      The flow of charges

           Ingredients

First, you need a conductor 

as a conduit for electricity

as a conduit for electricity

as a conduit for electricity

as a conduit for electricity

as a conduit for electricity

The Electric Current

DC-Circuits

The flow of electric charges

 DC-circuits

      The flow of charges

           The _Conventional_ Electric Current

Conventional current can be visualized as traffic or water flow

Conventional current always flows from high potential to low potential

The SI unit for conventional current is the Ampere

A = \frac{C}{s}

The Electric Current

The rate of charge flow in a conductor

I = \frac{dQ}{dt}

is defined as ...

 DC-circuits

      The flow of charges

           The _Conventional_ Electric Current

The SI unit for conventional current is the Ampere

A = \frac{C}{s}

The Electric Current

The rate of charge flow in a conductor

I = \frac{dQ}{dt}

is defined as ...

The DC Source

DC-Circuits

The flow of electric charges

 DC-circuits

      The flow of charges

           DC source

Electromotive Force (EMF)

A Direct-Current (DC) source* can be modeled as a two-terminal device that keeps one terminal (labeled +) at a fixed higher electric potential than the second terminal (labeled -)

*aka a source of Electromotive Force (EMF) ... a term that does not actually quantify a force, but rather a driving voltage (Electric Potential Difference.)

\mathcal E

+

-

The icon representing a DC source in a circuit. The long stick indicates the terminal at a higher potential.

 DC-circuits

      The flow of charges

           DC source

+

-

e^-

When a DC source is connected across an external electric load, the conduction electrons flow from the "negative" terminal through the load to reach the "positive" terminal.

Electrons "lose energy" as they go through the electric load; the electric potential energy gets converted into some other form. e.g. light or heat.

A battery is a source of EMF, providing the electrons with energy through chemical means.

 DC-circuits

      The flow of charges

           Boosting electrons with multiple sources

Sources connected in series boost the energy of the electrons sequentially resulting in an overall higher energy boost per electron.

Sources connected in parallel boost the energy of the electrons in tandem resulting in an overall higher rate of supply of electrons.

 DC-circuits

      The flow of charges

           Boosting electrons with multiple sources

Watch this video for an interesting visualization of the electric potential in a circuit with one or many DC sources.

Electric Resistance

DC-Circuits

The flow of electric charges

 DC-circuits

      The Electric Resistance

           Definition

The Electric Resistance is defined through Ohm's Law

R=\frac{V}{I}

Quantifies the restriction to current flow due to a potential difference.

\Omega={V}/{A}

SI Unit:

 DC-circuits

      The Electric Resistance

           of a solid conductor

The Electric Resistance of a conductor depends on the material and the geometric dimensions:

R=\frac{\rho\ L}{A}
Materials Resistivity
Conductors
Insulators

More in section 9.3: Resistance & Resistivity

10^{-8}\ \Omega m
10^{+8} - 10^{+14} \ \Omega m

For most common materials, the resistivity increases with increasing temperature:

\rho\approx{\rho_0}[1+\alpha(T-T_0)]

 DC-circuits

      The Electric Resistance

           Extra Resources

Basics of Electric Resistance

Length, Area, and Temperature

Electric Power

DC-Circuits

The flow of electric charges

 Electrostatics

      The Electric Potential

            relationship to the Electric Potential Energy

\Delta U=q\ \Delta V
q_0\ (V_{@P_2}-V_{@P_1})
\Delta U_\text{($q_0$ @ $P_1$)$\rightarrow$ ($q_0$ @ $P_2$)}=

The change in the Electric Potential Energy as some charge q0 is transferred from point P1 to point P2

The amount of net charge being transferred

The Electric Potential Difference between points P1 and P2

SI units: Substituting for

the charge in Coulombs, and

the Electric Potential in Volts,

results in the Energy in Joules

 DC-Circuits

      The Electric Power

            From the Electric Potential Energy

Power is defined as the rate of production or consumption of energy

P=\frac{\Delta U}{\Delta t}=\frac {q\ \Delta V}{\Delta t}=I\ \Delta V

Electric Power

For resistances (using Ohm's Law):

P=I\ \Delta V=I^2\ R=\frac{V^2}{R}

 DC-Circuits

      The Electric Power

            From the Electric Potential Energy

Suppose some number of electrons 

 DC-Circuits

      The Electric Power

            OpenStax

Capacitance

DC-Circuits

The flow of electric charges

 DC-circuits

      Capacitance

           Definition

If a potential difference         is created between two conductors separated by a small insulating gap, equal but opposite charges         will accumulate on the surfaces of the conductors across from each other. 

(V)
V
(\pm Q)
+Q
-Q

Capacitance is the ratio of the accumulated charge to the electric potential. 

C=\frac{Q}{V}

+

+

+

+

-

-

-

-

 DC-circuits

      Capacitance

           Parallel plate capactior

A parallel-plate capacitor has two identical conducting plates, each having a surface area A, separated by a distance d.

where \epsilon is the permittivity of the material in the space between the plates.

C=\frac{\epsilon A}{d}
d
V
A

 DC-circuits

      Capacitance

           Simulator

Adjust the plate area and separation and notice the effect on the capacitance.

 DC-circuits

      Capacitance

           Section 8.1

Equivalent Resistance

DC-Circuits

The flow of electric charges

 DC-circuits

      The Equivalent Resistance

           Extra Resources

Resistors in Series and Parallel

 DC-circuits

      Resistors in Series and Parallel

           TL;DR

Resistors are said to be in series whenever the current flows through the resistors sequentially.

Resistors are said to be in parallel whenever they are connected across the same potential difference.

 DC-circuits

      Resistors in Series and Parallel

           TL;DR

Series resistances add together to get the equivalent resistance:

The same current flows through each resistor in series.

The total potential drop across a series configuration of resistors is equal to the sum of the potential drops across each resistor.

R_s=R_1+R_2+... = \Sigma_iR_i

For series resistances:

 DC-circuits

      Resistors in Series and Parallel

           TL;DR

The equivalent resistance to a configuration of parallel resistances is given by the reciprocal of the sum of the reciprocals:

The potential drop across each resistor in parallel is the same.

The total current entering a parallel configuration of resistors is equal to the sum of the currents through each resistor.

R_p=\frac{1}{\tfrac{1}{R_1}+\tfrac{1}{R_2}+...} = \left(\Sigma_i\frac{1}{R_i}\right)^{-1}

For parallel resistances:

 DC-circuits

      Resistors in Series and Parallel

           TL;DR

For combinations of series and parallel resistances:

Find the equivalent resistance of all the resistors in the shown circuit:

Equivalent Capacitance

DC-Circuits

The flow of electric charges

 DC-circuits

      Capacitors in Series and Parallel

           TL;DR

Capacitors are said to be in series whenever they are connected sequentially.

Capacitors are said to be in parallel whenever they are connected across the same potential difference.

 DC-circuits

      Capacitors in Series and Parallel

           TL;DR

The equivalent capacitance is the sum of the capacitance

C_p=C_1+C_2+... = \Sigma_iC_i

Capacitors in Parallel:

Think of it as increasing the area of the plates.

 DC-circuits

      Capacitors in Series and Parallel

           TL;DR

The equivalent capacitance is the reciprocal of the sum of the reciprocals of the capacitance

Capacitors in Series:

C_s=\frac{1}{\tfrac{1}{C_1}+\tfrac{1}{C_2}+...} = \left(\Sigma_i\frac{1}{C_i}\right)^{-1}

Think of it as increasing the distance between the outer plates.

Kirchhoff's Rules

DC-Circuits

The flow of electric charges

 DC-circuits

      Kirchhoff's Rules

           Conservation of Current

Kirchhoff’s first rule—the junction rule: The sum of all currents entering a junction must equal the sum of all currents leaving the junction:

\Sigma I_\text{in}=\Sigma I_\text{out}

 DC-circuits

      Kirchhoff's Rules

           Conservation of Current

 DC-circuits

      Kirchhoff's Rules

           Conservation of Current

I_1=I_2+I_3
=
\Sigma I_\text{out}
\Sigma I_\text{in}

Kirchhoff's Junction Rule

must equal

The sum of all currents entering a junction

The sum of all currents leaving the junction.

 DC-circuits

      Kirchhoff's Rules

           Conservation of Current

\Sigma I_\text{in}

Kirchhoff's Junction Rule

must equal

The sum of all currents entering a junction

The sum of all currents leaving the junction.

=
\Sigma I_\text{out}

 DC-circuits

      Kirchhoff's Rules

           Conservation of Current

Kirchhoff’s first rule—the junction rule: The sum of all currents entering a junction must equal the sum of all currents leaving the junction:

\Sigma I_\text{in}=\Sigma I_\text{out}

 DC-circuits

      Kirchhoff's Rules

           Conservation of Current

 DC-circuits

      Kirchhoff's Rules

           The Loop Rule

 DC-circuits

           Extra Resources

Section 10.3

      Kirchhoff's Rules

 DC-circuits

      Kirchhoff's Rules

           Textbook

Section 10.3

Charging and Discharging a Capacitor

DC-Circuits

The flow of electric charges

  • A circuit containing a source of EMF, a resistor, an uncharged capacitor, and a switch.

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

  • Connect the switch to point A, which completes a circuit. The potential difference established by the battery drives a current that accumulates charge on the capacitor.
  • Connect the switch to point B, which completes a different circuit. The established potential difference across the capacitor drives a current through the resistor.
  • (a) circuit contains a source of EMF, a resistor, an uncharged capacitor, and a switch.

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

what is the charge on the capacitor?
what is the potential difference across the capacitor?
what is the potential difference across the resistor?
What is the current through the resistor (supplied by the EMF source) ?

@ before switch is connected to either A or B

0
0
0
0
  • (b) Connect the switch to (A) and start a timer (t=0)
what is the charge on the capacitor?
what is the potential difference across the capacitor?
what is the potential difference across the resistor?
What is the current through the resistor (supplied by the EMF source) ?

@ t=0 (immediately after switch is connected)

\mathcal{E}
0
0
\frac{\mathcal{E}}{R}

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

what is the charge on the capacitor?
what is the potential difference across the capacitor?
what is the potential difference across the resistor?
What is the current through the resistor (supplied by the EMF source) ?

@ t=t (some short time after switch ...)

\mathcal{E}-\frac{q(t)}{C}
\frac{q(t)}{C}
q(t)
\frac{\mathcal{E}}{R}-\frac{q(t)}{RC}

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

  • (b) Connect the switch to (A) and start a timer (t=0)
I=\frac{\mathcal{E}}{R}-\frac{q(t)}{RC}
\text{i.e.}\quad\frac{dq}{dt}=\frac{\mathcal{E}}{R}-\frac{q(t)}{RC}
q(t)=C{\mathcal{E}}\left(1-e^{-\frac{t}{RC}}\right)
\text{solve for}\quad q(t)

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

  • (b) Connect the switch to (A) and start a timer (t=0)

@just after

Capacitor carries no charge

@long_time

Capacitor carries maximum charge

q(t)=C{\mathcal{E}}\left(1-e^{-\frac{t}{RC}}\right)
what is the charge on the capacitor?
what is the potential difference across the capacitor?
what is the potential difference across the resistor?
What is the current through the resistor (supplied by the EMF source) ?
C{\mathcal{E}}
{\mathcal{E}}
0
0

@ t=       (long time after switch ...)

\infty

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

  • (b) Connect the switch to (A) and start a timer (t=0)
I=\frac{dq}{dt}
=\frac{d}{dt}C{\mathcal{E}}\left(1-e^{-\frac{t}{RC}}\right)
=\frac{\mathcal{E}}{R}\ e^{-\frac{t}{RC}}
=I_0\ e^{-\frac{t}{\tau}}

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

  • (b) Connect the switch to (A) and start a timer (t=0)

@just after

Capacitor draws maximum current

@long_time

Capacitor draws

no current

q(t)=C{\mathcal{E}}\left(1-e^{-\frac{t}{RC}}\right)
I(t)=I_0\ e^{-\frac{t}{\tau}}

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series


 
(a) The switch has been open for a long time
(b) Just after the switch is closed
(c) A long time after the switch has been closed
I_1
I_2
I_3

Check your understanding

0
0
0
\frac{V}{R_1}
0
\frac{V}{R_1}
\frac{V}{R_1+R_2}
\frac{V}{R_1+R_2}
0

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

  • (b) Connect the switch to (B) and start a new timer (t=0)

@t=0 (immediately after switch is connected to B)

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

what is the charge on the capacitor?
what is the potential difference across the capacitor?
what is the potential difference across the resistor?
What is the current through the resistor (supplied by the EMF source) ?
C{\mathcal{E}}
{\mathcal{E}}
\frac{{\mathcal{E}}}{R}
{\mathcal{E}}
  • (b) Connect the switch to (B) and start a new timer (t=0)

@t=0 (immediately after switch is connected to B)

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

what is the charge on the capacitor?
what is the potential difference across the capacitor?
what is the potential difference across the resistor?
What is the current through the resistor (supplied by the EMF source) ?
C{\mathcal{E}}
{\mathcal{E}}
\frac{{\mathcal{E}}}{R}
{\mathcal{E}}
  • (b) Connect the switch to (B) and start a new timer (t=0)

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

what is the charge on the capacitor?
what is the potential difference across the capacitor?
what is the potential difference across the resistor?
What is the current through the resistor (supplied by the EMF source) ?

@ t=t (some short time after switch ...)

\frac{q(t)}{C}
\frac{q(t)}{C}
q(t)
\frac{q(t)}{RC}
  • (b) Connect the switch to (B) and start a new timer (t=0)

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

I=-\frac{q(t)}{RC}
\text{i.e.}\quad\frac{dq}{dt}=-\frac{q(t)}{RC}
q(t)=C{\mathcal{E}}\left(e^{-\frac{t}{RC}}\right)
\text{solve for}\quad q(t)

@just after

Capacitor carries no charge

@long_time

Capacitor carries maximum charge

q(t)=C{\mathcal{E}}\left(e^{-\frac{t}{RC}}\right)
I(t)=-I_0\ e^{-\frac{t}{\tau}}

 DC-circuits

      Charging/Discharging a Capacitor

           RC in series

 DC-circuits

           Simulator

Use the sliders to adjust the battery voltage, the resistor's resistance, the plate area, and the plate separation. Use the check boxes to open and close the switch, as well as turn the animation on one off. When animation is turned off, you can use the step buttons to advance time forward or backward in small steps.

      Charging/Discharging a Capacitor