Electric current and dc circuits.

Physics 201
Professor P. Q. Hung
311B, Physics Building
Physics 201 – p. 1/3

Electric Current and DC Circuits
Summary of last lecture
Equipotential surfaces: Surfaces where the
potential is the same everywhere, e.g. the
surface of a conductor.

Q = C|∆V |. C: Capacitance, the capacity to
store charge

Q = C|∆V |. C: Capacitance, the capacity to
store charge
U = 1
2QV = 1
2CV 2
= Q2
2C : potential energy of a
capacitor

Beyond electrostatic
A current of 0.2 mA coming from a 3.0 V battery
operates a calculator for one hour. How much
charge flows in the circuit?
In previous lectures, a conductor in
electrostatic equilibrium: No electric field
inside ⇒ Conduction electrons do not flow.

For conduction electrons to start ﬂowing
together (current) in a given direction, we
need an electric ﬁeld.

For conduction electrons to start flowing
together (current) in a given direction, we
need an electric field.
Difference in potential ⇒ Electric field.

Electric Current
Difference in potential ⇒ Electric ﬁeld ⇒
Conduction electrons move.

Electric Current
Difference in potential ⇒ Electric ﬁeld ⇒
Conduction electrons move.
How do we create such a potential
difference? By connecting the two ends of the
wire to the two terminals of a battery which
posesses an electric potential difference.

Electric Current
How does a battery create a potential
difference between its two terminals? By
chemical reactions which transfer electrons
from one terminal (making it positively
charged) ( higher potential) to the other
terminal (making it negatively charged) (lower
potential).

Electric Current

Electric Current
Is there a limit to the potential difference
between the two terminals of a battery? Yes.
It is called the electromotive force E (nothing
to do with a force), e.g. E = 1.5 V for a AA
battery.

Electric Current
battery.
How do the conduction electrons move?
From low to high potential i.e. from - to +.

Electric Current
battery.
How do the conduction electrons move?
From low to high potential i.e. from - to +.
(Historical) convention: The direction of the
current is taken to be from + to -, opposite to
the direction of the electrons.

Electric Current

Electric Current
How many electrons pass through a cross
section of the wire in one second?

Electric Current
Current:
I = ∆ q
∆ t

Electric Current
Current:
I = ∆ q
∆ t
Unit: 1 ampere(A) = 1 C/s

Electric Current: Example
Answer:
∆ q = I∆ t = (0.2 × 10−3
A)(3600 s) = 0.72 C
Physics 201 – p. 10/3

Electric current
If the current is always in the same direction,
you have a direct current or dc current; If the
current oscillates, i.e. changes direction, you
have an alternating or ac current.
Physics 201 – p. 11/3

Electric current
If the current is always in the same direction,
you have a direct current or dc current; If the
current oscillates, i.e. changes direction, you
have an alternating or ac current.
What is a typical speed of the electrons in a
current?
Answer: A rough calculation indicates that the
average speed of the electrons called the drift
speed is around 2.4 × 10−4
m/s.
Physics 201 – p. 11/3

Electric current
So if I have a wire of length 2.4m, an electron
at one end will take 10, 000 s to reach the
other end. Why is it that when I ﬂip the switch,
the light immediately turns on?
Answer: Just because the signal that turns on
the electric ﬁeld travels at the speed of light
so that all electrons from one end to the other
move at once.
Physics 201 – p. 12/3

Ohm’s Law
How much current is ﬂowing inside a circuit
hooked to a battery? Take a ride on one of
these electrons. You can actually see that it
collides repeatedly with the atoms of the wire
⇒ Resistance to the motion of that electron.
Physics 201 – p. 13/3

Ohm’s Law
Any similarity with something that we already
know? Yes. Imagine that you are sliding down
a very icy slope. Because of negligible
friction, most of the potential energy is
converted into kinetic energy. If the slope is
very rough instead, some of that potential
energy is converted into heat.
Physics 201 – p. 14/3

Ohm’s Law
The resistance is translated into a
relationship between the applied voltage V
and the current I: Ohm’s Law:
R = V
I
Physics 201 – p. 15/3

Ohm’s Law
The resistance is translated into a
relationship between the applied voltage V
and the current I: Ohm’s Law:
R = V
I
R: resistance. Unit: 1 ohm(Ω) = 1 V
A
Physics 201 – p. 15/3

Ohm’s Law: Example
The resistance of a bagel toaster is 14 Ω. To
prepare a bagel, the toaster is operated for one
minute from a 120-V outlet. How much energy is
delivered to the toaster?
Three inputs: R = 14 Ω; t = 60 s; V = 120 V .
Concepts?
Physics 201 – p. 16/3

The resistance of a bagel toaster is 14 Ω. To
prepare a bagel, the toaster is operated for one
minute from a 120-V outlet. How much energy is
delivered to the toaster?
Three inputs: R = 14 Ω; t = 60 s; V = 120 V .
Concepts?
The energy delivered is equal to the work
done in moving ∆q in ∆t = 60s and by a
potential difference of 120 V. ⇒ E = (∆ q) V .
Physics 201 – p. 16/3

What’s ∆q?
∆q = I∆ t = V
R ∆ t
⇒ E = (∆ q) V = V 2
R ∆ t = (120V )2
14Ω (60s) =
6.2 × 104
J.
Physics 201 – p. 17/3

Resistance and Resistivity
When I am given a piece of conducting wire,
how do I know what its resistance might be?
Answer: The electrons that travel from one
end to the other encounter more atoms to
scatter on as the wire gets longer. Also if the
atoms are packed into a smaller area, there
will be more scatterings ⇒ The resistance will
get larger.
Physics 201 – p. 18/3

When I am given a piece of conducting wire,
how do I know what its resistance might be?
Answer: The electrons that travel from one
end to the other encounter more atoms to
scatter on as the wire gets longer. Also if the
atoms are packed into a smaller area, there
will be more scatterings ⇒ The resistance will
get larger.
So?
Physics 201 – p. 18/3

Empirical formula for the resistance:
R = ρL
A
ρ: Resistivity of the material
L: Length of conducting wire
A: Its cross-section
Physics 201 – p. 19/3

Empirical formula for the resistance:
R = ρL
A
ρ: Resistivity of the material
L: Length of conducting wire
A: Its cross-section
What does that tell us about different
material? Conductors have low resistivity,
while insulators have large resistivity. In
general, we want to minimize the resistance.
Physics 201 – p. 19/3

Physics 201 – p. 20/3

Resistance and Resistivity: Some aplications
Impedance (or resistance) plethysmography:
Measure the resistance in the calf,
R = ρL
A = ρ L
Vcalf /L = ρ L2
Vcalf
. Pressure cuff cuts
off the veinous ﬂow ⇒ Vcalf increases ⇒ R
decreases. Pressure cuff ⇒ removed ⇒
Rapid return to normal resistance if there is
no blood clot in the veins. A slow return to
normal indicates some blood clot.
Physics 201 – p. 21/3

20-gauge wire’s cross section: 5.2 × 10−7
m2
;
16-gauge wire’s cross section: 13 × 10−7
m2
.
For the same length, the 16-gauge wire has
smaller resistance than the 20-gauge one ⇒
less heating (proportional to R) of the wire.
Physics 201 – p. 22/3

If I heat up a wire, will its resistance change?
Answer: The resistance goes up! For
example, R = R0(1 + α(T − T0)) where α is
the temperature coefﬁcient of resistivity.
Physics 201 – p. 23/3

If I cool the wire to extremely low
temperatures, what will happen to its
resistance?
Answer: There are some material whose
resistance goes to zero as the temperature is
lowered below some critical temperature Tc.
They are called superconductors. Copper
oxide complexes such as
Hg − Ba2Ca2Cu2O8+δ have Tc = 150 K.
Physics 201 – p. 24/3

Electrical energy and power
From the example given above, the energy
delivered to the toaster is
∆U = ∆qV = IV ∆t
Physics 201 – p. 25/3

From the example given above, the energy
delivered to the toaster is
∆U = ∆qV = IV ∆t
The power is
P = ∆U
∆t = IV = I2
R = V 2
R
Physics 201 – p. 25/3

For the toaster example above, P = 1.03kW
Physics 201 – p. 26/3

For the toaster example above, P = 1.03kW
From Eq. (5), one can see that, in order to
minimize the power dissipated in terms of
heat, one has to minimize the resistance.
Physics 201 – p. 26/3

Resistors in series
What happens to a circuit when I connect
resistors in series, i.e. one after the other?
Answer: In series means that the same
current ﬂows through the resistors.
Physics 201 – p. 27/3

Resistors in series
Let me take two resistors, R1 and R2. Can I
simplify the problem ?
Answer: Yes. The voltages across the
resistors are respectively V1 = IR1 and
V2 = IR2. The sum is equal to the emf of the
battery (neglecting internal resistance of the
battery):
V = V1 + V2 = IR1 + IR2 = I(R1 + R2) = IReq.
Equivalent resistance:
Req = R1 + R2 + ...
Physics 201 – p. 28/3

Resistors in series
So does that tell me that for a circuit with
resistors in series, I can draw an equivalent
circuit with one resistor whose resistance is
the sum of all the individual resistances?
Answer: Yes!
Physics 201 – p. 29/3

Resistors in series
Physics 201 – p. 30/3

Resistors in parallel
How come all the wall sockets in my house
have the same voltage, namely 120 V?
Answer: This is an example of a wiring in
parallel.
Physics 201 – p. 31/3

How come all the wall sockets in my house
have the same voltage, namely 120 V?
Answer: This is an example of a wiring in
parallel.
What does it really mean?
Answer: In parallel means that the devices
(resistors, etc..) are connected in such a way
that the voltage across each one of them is
the same.
Physics 201 – p. 31/3

What about the current(s)?
Answer: Since I = V/R and V is the same,
the one with larger R will have a smaller
current ﬂowing in it. There will be a current Ii
ﬂowing in each branch i. The sum of all the
currents in all the branches should be equal
to the current produced by the source
(battery,etc..)
I = I1 + I2 + I3 + ...
Physics 201 – p. 32/3

Can I draw an equivalent circuit?
Answer: Yes.
I = I1 + I2 + I3 + ... = V
R1
+ V
R2
+ .. = V
Req
1
Req
= 1
R1
+ 1
R2
+ ..
Physics 201 – p. 33/3

Physics 201 – p. 34/3

Electric current and dc circuits.

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