Download Direct Current Circuits

Lecture 6
Direct Current Circuits
Chapter 18
Outline
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Energy Source in Circuits
Resistor Combinations
Kirchhoff’s Rules
RC Circuits
Energy in Circuits
Direct current is a current flowing in one direction
through a circuit.
Circuits are usually powered by batteries or
generators, which are called sources of emf.
The emf (ℇ)stands for electromotive force and is
not a fundamental force.
ℇ is the work per unit charge and is measured in
volts (V).
ℇ equals to the potential difference across the
battery if the battery’s internal resistance is
neglected.
Emf and Circuit Parameters
V = ℇ  IRint
V = IRload
ℇ = IRload
+ IRint
ℇ
I = 
Rload + Rint
Emf
ℇ is equal to the terminal voltage in the absence of
the current.
ℇI = I2Rload +I2Rint
The total power output of the source of emf is
converted to:
• the power delivered to the load resistance +
• the power delivered to the internal resistance
In all the problems, we assume that the internal
resistance of the battery is negligible.
Resistors in Series
The current (I) is the same
in both resistors.
V1 = IR1, V2=IR2
V = IR1 + IR2 = I(R1+R2)
V = IReq
Req = R1 + R2
Example
The equivalent resistance of a series combination
of resistors equals to the algebraic sum of the
individual resistances.
Resistors in Parallel
The potential differences
across both resistors are
the same (V), as each is
directly connected to the
battery.
Due to conservation of
charge, Itot = I1 + I2
V
V
V
1 1 1
Itot =  = I1 + I2 =  +    =  + 
Req
R1
R2
Req R1
Kirchhoff’s Rules
Circuit elements can be connected in various ways.
For analysis of complex circuits, 2 rules proposed
by Gustav Kirchhoff in 1845 can be applied.
1. The sum of all the currents entering any junction
point is equal to the sum of all the currents
leaving that junction point (junction rule).
2. The sum of the potential differences across all
the elements around any closed-circuit loop
must be zero (loop rule).
Kirchhoff’s rules
The junction rule is a consequence of conservation
of charge, the loop rule is a consequence of
conservation of energy.
Application of Kirchhoff’s rules
RC Circuits
Circuits which contain capacitors have timedependent currents.
Q = Cℇ
Q and ℇ are the maximum charge
and voltage across the capacitor
C is the capacitance
q(t) = Q(1  et/RC)
e = 2.718
 = RC is the capacitor’s
time constant
RC Circuits
Another view of a RC circuit
Problem: An uncharged capacitor and a resistor
are connected in a series to an emf source.
ℇ = 10 V, C = 20 F, R = 100 .
Find: the circuit’s time constant,
the maximum charge on the capacitor,
the charge on the capacitor after 1 time constant.
Solution:  = RC
= (102 ) (2 105 F) = 2 103 c
Q = Cℇ = (2 105 F) (10 V) = 2 104 C = 200 C
q = 0.632 Q = 0.632 (200 C) = 126 C
Summary
• Direct current is current that flow in a circuit in
one direction
• Kirchhoff’s rules are a set of instructions to
analyze complex circuits. The rules are
consequences of conservation of charge and
energy.
• Circuits containing capacitors have timedependent currents. The charge on the capacitor
and time dependence of the current through it
can be determined using the capacitor time
constant.