Answer:
Explanation: Draw a circuit with resistors in parallel and in series.
Calculate the voltage drop of a current across a resistor using Ohm’s law.
Contrast the way total resistance is calculated for resistors in series and in parallel.
Explain why total resistance of a parallel circuit is less than the smallest resistance of any of the resistors in that circuit.
Calculate total resistance of a circuit that contains a mixture of resistors connected in series and in parallel.
According to Ohm’s law, the voltage drop,
V
, across a resistor when a current flows through it is calculated using the equation
V
=
I
R
, where
I
equals the current in amps (A) and
R
is the resistance in ohms
(
Ω
)
. Another way to think of this is that
V
is the voltage necessary to make a current
I
flow through a resistance
R
.
So the voltage drop across
R
1
is
V
1
=
I
R
1
, that across
R
2
is
V
2
=
I
R
2
, and that across
R
3
is
V
3
=
I
R
3
. The sum of these voltages equals the voltage output of the source; that is,
V
=
V
1
+
V
2
+
V
3
.
This equation is based on the conservation of energy and conservation of charge. Electrical potential energy can be described by the equation
P
E
=
q
V
, where
q
is the electric charge and
V
is the voltage. Thus the energy supplied by the source is
q
V
, while that dissipated by the resistors is
q
V
1
+
q
V
2
+
q
V
3
.
Connections: Conservation Laws
The derivations of the expressions for series and parallel resistance are based on the laws of conservation of energy and conservation of charge, which state that total charge and total energy are constant in any process. These two laws are directly involved in all electrical phenomena and will be invoked repeatedly to explain both specific effects and the general behavior of electricity.
These energies must be equal, because there is no other source and no other destination for energy in the circuit. Thus,
q
V
=
q
V
1
+
q
V
2
+
q
V
3
. The charge
q
cancels, yielding
V
=
V
1
+
V
2
+
V
3
, as stated. (Note that the same amount of charge passes through the battery and each resistor in a given amount of time, since there is no capacitance to store charge, there is no place for charge to leak, and charge is conserved.)
Now substituting the values for the individual voltages gives
V
=
I
R
1
+
I
R
2
+
I
R
3
=
I
(
R
1
+
R
2
+
R
3
)
.
Note that for the equivalent single series resistance
R
s
, we have
V
=
I
R
s
.
This implies that the total or equivalent series resistance
R
s
of three resistors is
R
s
=
R
1
+
R
2
+
R
3
.
This logic is valid in general for any number of resistors in series; thus, the total resistance
R
s
of a series connection is
$
R
s
=
R
1
+
R
2
+
R
3
+
…