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Unformatted text preview: Figure 23: Three Resistors in Series with
Their Simpliﬁed Equivalent Resistor Figure 24: Three Resistors in Parallel with
Their Simpliﬁed Equivalent Resistor Next, we pass through the three resistors at each of which we lose iRi in potential. After that, we
ﬁnd ourselves back at point b and therefore
V − iR1 − iR2 − iR3 = 0 ⇒ V = iR1 + iR2 + iR3 ⇒ V = i (R1 + R2 + R3 ) By comparing the last form of the expression with Ohm’s law, we can see what the resistance of
our simpliﬁed resistor is. For n resistors in series, the equivalent simpliﬁed resistor is
n Req = Rj (26) j =1 We now want to do the same thing, except for multiple resistors in parallel (see ﬁgure 24). In
this case, we will use the fact that since opposite sides of each resistor are wired to opposite sides
of the battery, the potential diﬀerence across each resistor must be the same. We will also use the
fact noticed above that the sum of the currents through each resistor must be equal to the current
ﬂowing through the rest of the circuit (the battery in this case).
i = i1 + i2 + i3 = V
V
V
+
+
=V
R1 R2 R3 1
1
1
+
+
R1 R2 R3 Or, rearranging
V =i 1
1
R1 + 1
R2 + 1
R3 Hence, we see that (similar to capacitors in series) n resistors in parallel can be reduced to an
equivalent resistor of resistance
n
1
1
=
(27)
R eq j =1 R j 8.4 RC Circuits Up until now we have dealt with circuits which involve either capacitors of resistors but none which
involve both at the same time. Lets consider this possibility now. The two situations which we will
consider are: 1. The capacitor begins charged and is then allowed to discharge through a resistor.
2. The capacitor begins uncharged and is then charged by a battery through a resistor. Figure 25
illustrates how we could see both of these situations in one circuit. When the switch is in position a
then the capacitor is charging through the resistor. When the switch is in position b the capacitor
is discharging through the resistor.
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This note was uploaded on 12/05/2011 for the course PHY 2049 taught by Professor Any during the Spring '08 term at University of Florida.
 Spring '08
 Any
 Physics

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