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Kirchhoff’s First Rule (“Loop Rule” or
“Kirchhoff’s Voltage Law”)
The algebraic sum of the changes in potential encountered
in a complete traversal of any loop of circuit must be zero.
Move around circuit:
I R 1 R 2
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Rules^ ^1
I^ R^1 R^2
Voltage Gains enter with a + sign. Voltage Drops enter with a - sign.
Note: always points from negative to positve
a b - +
b a I a b I b a 8/16/12 3
Loop Example
a d b c e f
R 1
I
R 2 R 3
R 4
I
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Internal Resistance of an emf Device
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Resistors in Series
^1^ I
!
-^
I
I
*When a potential difference, V, is applied across resistances in series, the resistances have identical current.
- Resistances in series can be replaced with an equivalent resistance, REQ, that has the same current I and the same total potential difference V as the actual resistances. 8/16/12 6
Another (intuitive) way
Consider two cylindrical resistors with lengths L 1 and L 2
V
R 1
R 2
L 2 L 1 Put them together, end to end to make a longer one... R 1 = ρ L 1 A R 2 = ρ L 2 A Req = ρ
L 1 + L 2
A
= R 1 + R 2
R = R 1 + R 2
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Kirchhoff’s Second Rule (Junction Rule or
“Kirchhoff’s Current Law”)
The sum of the current entering any junction must be equal
to the sum of the currents leaving that junction.
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How to use Kirchhoff’s Law:
Analyze the circuit & identify all circuit nodes: Identify all independent loops & use Kirchhoff’s Voltage Law:
R (^3) R^2 R 1 a b c d
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- When you are given a circuit, you must first carefully analyze circuit topology. - find the nodes and distinct branches and pick Linearly Independent subsets of each. - assign branch currents
- Use Kirchhoff’s First Rule for all independent loops in the circuit.
- Use Kirchhoff’s Second Rule for all independent nodes in circuit.
Problem Solving Tips
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Example 25-
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Cramer’s Rule
If
then
a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2 x = c 1 b 1 c 2 b 2 a 1 b 1 a 2 b 2
c 1 b 2 − c 2 b 1 a 1 b 2 − a 2 b 1 y = a 1 c 1 a 2 c 2 a 1 b 1 a 2 b 2
a 1 c 2 − a 2 c 1 a 1 b 2 − a 2 b 1
