CURRENT AND CIRCUITSAP Physics C: E&M

INTRODUCTORY TERMSCurrent: Charge Flow. This is the drift of

electrons due to a potential difference.

AC: Alternating current. The polarity of the voltage source switches back and forth causing charges in

path to vibrate.DC: Direct current. A constantly applied

voltage causes charged particles to drift in one direction

Series: Elements in circuit are connected along one path.

Parallel: Elements of circuits are connected on separate branches.

CAPACITORS IN A DC CIRCUIT

+-

Voltage source

C1

C2

C3

Adding capacitors in series will lower the capacitance of the circuit when compared to the possible

capacitance of just one capacitor in the circuit.

Only the first plate of the first capacitor and the last plate of the last capacitor are actually connected to the voltage source, so only these plates will gain or lose electrons due to the potential difference of the

battery.

CAPACITORS IN A DC CIRCUIT

+-

Voltage source

C1

C2

C3

The inner plates are induced with charge. All capacitors carry an equivalent charge Q.

The voltage across all elements in the series will add up to that of the battery. Each capacitor has a different

capacitance and has the same charge, so the individual voltages will differ.

1 2 ... nV V V V

CAPACITORS IN A DC CIRCUIT

+-

Voltage source

C1

C2

C3

This should not be surprising since you are basically just making one big capacitor with a larger separation

(d).

Q is the same for all so the equivalent capacitance can be found with:

21

...eq n

Q Q Q Q

C C C C

21

1 1 1 1...

eq nC C C C

CAPACITORS IN A DC CIRCUIT

+-

Voltage source

C1

Adding capacitors in parallel will raise the capacitance of the circuit when compared to the possible

capacitance of just one capacitor in the circuit.

All capacitors are directly connected to the same voltage source so they will each reach the same

potential difference when charged.

C2 C3

CAPACITORS IN A DC CIRCUIT

Since each capacitor may have a different capacitance, each may hold a different amount of

charge, but the sum of the charge will equal that of one capacitor to replace those in parallel.

1 2 ... nQ Q Q Q

+-

Voltage source

C1 C2 C3

CAPACITORS IN A DC CIRCUIT

This should not be surprising since you are basically just making one big capacitor with a larger surface

area (A) for charge to be stored.

V is the same for all so the equivalent capacitance can be found with:

1 2 ...eq nC V CV C V C V

+-

Voltage source

C1 C2 C3

1 2 ...eq nC C C C

PRACTICE PROBLEMS #’S 8-12

CIRCUIT COMPONENTS

+-

+-

+-

+-

+- A

+-

B

C

+-

D E

ELECTRIC CURRENT

It takes over 6.24 billion billion electrons to add

up to one coulomb!

1 C of charge through any cross section of wire per second is one AMP!

Electric current is the amount of charge passing through a certain area per second. It is measured in amperes.

ELECTRIC CURRENT

Iav Qt

If the charge flow rate varies, we define the

instantaneous current as:

IdQ

dt

The direction of current is the direction that positive charges would flow if free to do so.

n=number of charge carriers per unit volume A=cross-sectional area of wireΔx=length of section of wireΔQ=charge in a section of wireq=charge on each particle

Q nAx q

If charge carriers move with a velocity vd, then they move a distance Δx=vdΔt

ELECTRIC CURRENT

QnAv dtq

IQt

nAv dq

With no voltage, charges in a metal bounce around randomly similar to gas

molecules. With a voltage they still bounce around but slowly drift in one

direction.

DRIFT VELOCITY

A copper wire with cross-sectional area3x10-

6m2 carries a current of 10.0A. Find the drift speed of the electrons. The density of copper

is 8.95g/cm3.

DRIFT VELOCITY

from the periodic tableatomic mass of copper:

m=63.5g/mol

Vm

63.5g /mol

8.95g /cm3 7.09cm3 /mol

A copper wire with cross-sectional area3x10-

6m2 carries a current of 10.0A. Find the drift speed of the electrons. The density of

copper is 8.95g/cm3.

DRIFT VELOCITY

nnAV

6.021023electrons/mol

7.09cm3 /mol8.48x1022electrons/cm3

vd I

nqA

10A

8.48x1028electons/m3 1.6x10 19C 3x10 6m2

vd 2.46x10 4m/s

THEN HOW DO THE LIGHTS COME ON SO FAST?

CURRENT DENSITY

We will define current density as:

JI

Anqvd

A current density J and an electric field E are established in a conductor when a

potential difference is maintained across the conductor.

The proportionality constant is called the conductivity of the conductor.

JE

OHM’S LAW

Named after Georg Simon Ohm (1787-1854)

For many materials, the ratio of the current density to the electric field is a constant,

(sigma), that is independent of the electric field producing the current.

If the potential difference is constant, the current is constant.

This is not a law of nature, but an empirical relationship found to be valid for certain

materials (most metals)

OHM’S LAW

For a segment of wire of length L:

VEL

JV

L

JE

VJL

IL

A

R V

IL

AResistance!

RESISTANCE

The unit is the Ohm (Ω)

1 1V

1A

1

The inverse of conductivity is resistivity!

R LA

RESISTANCE AND TEMPERATURE:

0 1 T T0

R R0 1 T T0

For all metals, resistivity increases with temperature

increase.

some reference value usually at

20°C

Temperature coefficient of

resistivity

ELECTRICAL ENERGY AND POWER

UqVDivide both sides by time.

U

tqV

t

PIV

ELECTRICAL ENERGY AND POWER

VIR

PI2R

PIV

IV

R

PV

R

2

R

PV2

R

ELECTROMOTIVE “FORCE” – (EMF)An emf is any device (generator/battery)

that produces an electric field and thus may cause charges to move around in a circuit.

Is an emf (ε) any different than a voltage source (V)?

Any real emf has a certain amount of its own internal resistance, so the voltage that it will

supply to a circuit between terminals is slightly different than its own potential

difference.Both are measured in Volts.

ELECTROMOTIVE “FORCE” – (EMF)An emf can be thought of as a charge pump.

V Ir V is the terminal voltageEpsilon is the potential difference of the emfI is the circuit’s currentr is the internal resistance of the emfR is the equivalent resistance of the circuitP is the power dissipated in circuit and emf device

VIR

IR Ir

IR Ir

PI I2R I2r

KIRCHOFF’S RULES FOR COMPLEX CIRCUITS:

I am Bunsen. Have you tried my

burner?

The sum of the currents entering any junction must equal

the sum of the currents leaving that

junction.

The algebraic sum of the changes in

potential across all of the elements around any closed loop must

be zero.

KIRCHOFF’S RULES FOR COMPLEX CIRCUITS:

Do you mean that Energy and Charge are conserved?

Of course Bunsen,If charge is split

between two branches it must flow down one path. it will not build up in a location or

disappear.

Also, a charge must gain as much energy as it loses throughout the circuit because it begins and ends at

the same point.

By the way, nice burner!

RC CIRCUITSWhat is different about a circuit with a

resistor and a capacitor than one with just a resistor?

The current does not flow at a constant rate!

Why is this?

+-

The charge stops flowing when a capacitor matches the battery

voltage. It drains charge through the resistor after batter is

disconnected.

+++

-- - -

++++++

- - - - - - - -

C C RR

No current I

ΔVR=0 ΔVR=-IR

ΔVC=Q0/C ΔVC=Q/C

At time t=0 the switch is closed and the full capacitor discharges.

From the loop rule…

VC VR 0

Q

C IR 0

Q

C IR 0

IdQ

dt

Q

CdQ

dtR 0

Q and I are instantaneous values:

dQ

dt

Q

RC

dQ

Q

dt

RC

dQ

QQ0

Q

1

RCdt

0

t

lnQ

Q0

t

RC

eln

Q

Q0

e

t

RC

Q

Q0

et

RC

FIND THE CURRENT EXPRESSION FOR AN RC CIRCUIT

IdQ

dt

QQ0et

RC