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Direct Current Circuits:
3-1 EMF
3-2 Resistance in series and parallel .
3-3 Rc circuit
3-4 Electrical instruments
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Example:
 (A) Find the current in the circuit.
starting at a, we see that a b
represents a potential difference
of + Ɛ1
b  c represents a potential
difference of -IR1,
c  d represents a potential
difference of - Ɛ2, and
d  a represents a potential
difference of -IR2
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The negative sign for I indicates that the direction of the current is
opposite the assumed direction.
(B) What power is delivered to each resistor? What
power is delivered by the 12-V battery?
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Problem-Solving Strategy – Kirchhoff’s Rules
 Draw the circuit diagram and assign labels and
symbols to all known and unknown quantities
 Assign directions to the currents.
 Apply the junction rule to any junction in the
circuit
 Apply the loop rule to as many loops as are
needed to solve for the unknowns
 Solve the equations simultaneously for the
unknown quantities
 Check your answers
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Example:
A) Under steady-state conditions, find the unknown currents
I1, I2, and I3 in the multiloop circuit shown in Figurev
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(B) What is the charge on the capacitor?
We can apply Kirchhoff’s loop rule to loop bghab
(or any other loop that contains the capacitor) to find the
potential difference ∆Vcap across the capacitor. We use
this potential difference in the loop equation without
reference to a sign convention because the charge on
the capacitor depends only on the magnitude of the
potential difference.
Moving clockwise around this loop, we obtain
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Quiz 3:
 In using Kirchhoff’s rules, you generally assign a
separate unknown current to
 (a) each resistor in the circuit (b) each loop in the
circuit (c) each branch in the circuit (d) each battery
in the circuit.
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Ans.
 (c) each branch in the circuit
A current is assigned to a given branch of a circuit.
There may be multiple resistors and batteries in a given
branch.
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RC Circuits
 A direct current circuit may contain capacitors and
resistors, the current will vary with time
 When the circuit is completed, the capacitor starts
to charge
 The capacitor continues to charge until it reaches
its maximum charge (Q = Cε)
 Once the capacitor is fully charged, the current in
the circuit is zero
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Charging Capacitor in an RC Circuit
 The charge on the capacitor
varies with time
 q = Q(1 – e-t/RC)
 The time constant, =RC
 The time constant represents
the time required for the charge
to increase from zero to 63.2%
of its maximum
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The following dimensional analysis shows that τ has the
units of time:
The energy output of the battery as the capacitor is fully
charged is Q Ɛ= Ɛ C 2.
After the capacitor is fully charged, the energy stored in
the capacitor is 1/2Q Ɛ = 1/2 CƐ 2, which is just half the
energy output of the battery.
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Notes on Time Constant
 In a circuit with a large time constant, the
capacitor charges very slowly
 The capacitor charges very quickly if there is a
small time constant
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Discharging Capacitor in an RC Circuit
 When a charged capacitor is
placed in the circuit, it can be
discharged
q = Q e-t/RC
 The charge decreases
exponentially
 At t =  = RC, the charge
decreases to 0.368 Qmax
 In other words, in one time
constant, the capacitor loses
63.2% of its initial charge
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5- Electric meter
Ammeters and Voltmeters
An ammeter is a device for measuring current, and a
voltmeter measures voltages.
The current in the circuit must flow through the ammeter;
therefore the ammeter should have as low a resistance as
possible, for the least disturbance.
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Ammeters and Voltmeters
A voltmeter measures the potential
drop between two points in a circuit. It
therefore is connected in parallel; in
order to minimize the effect on the
circuit, it should have as large a
resistance as possible.
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Summary
• Power in an electric circuit:
• If the material obeys Ohm’s law,
• Energy equivalent of one kilowatt-hour:
• Equivalent resistance for resistors in series:
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Summary
• Inverse of the equivalent resistance of resistors in series:
• Junction rule: All current that enters a junction must
also leave it.
• Loop rule: The algebraic sum of all potential charges
around a closed loop must be zero.
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Summary
• Equivalent capacitance of capacitors connected in parallel:
• Inverse of the equivalent capacitance of capacitors
connected in series:
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Kirchhoff’s rules:
1- Junction rule. The sum of the currents entering any
junction in an electric circuit must equal the sum of the
currents leaving that junction:
2- Loop rule. The sum of the potential differences across all
elements around any circuit loop must be zero:
The first rule is a statement of conservation of charge; the
second is equivalent to a statement of conservation of energy.
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Note:
When a resistor is traversed in the direction of the current,
the potential difference ∆V across the resistor is -IR.
When a resistor is traversed in the direction opposite the
current, ∆V = +IR.
When a source of emf is traversed in the direction of the emf
(negative terminal to positive terminal), the potential
difference is +Ɛ .
When a source of emf is traversed opposite the emf (positive
to negative), the potential difference is -Ɛ
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• Charging a capacitor:
where Q = C Ɛ is the maximum charge on the capacitor. The
product RC is called the time constant Ƭ of the circuit.
• Discharging a capacitor:
where Q is the initial charge on the capacitor and I0 = Q /RC is
the initial current in the circuit.
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Summary
• Ammeter: measures current. Is connected in series.
Resistance should be as small as possible.
• Voltmeter: measures voltage. Is connected in parallel.
Resistance should be as large as possible.
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Household Circuits
 The utility company
distributes electric power
to individual houses with
a pair of wires
 Electrical devices in the
house are connected in
parallel with those wires
 The potential difference
between the wires is
about 120V
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