13.7.4 Solving Circuit Problems

When wires join voltage sources and resistors together, electric field permeates through the wires across the resistors. Electrical forces send electrons tumbling through the metal lattice. But have you noticed that we do not solve circuits by calculating electric field and forces? Instead we solve circuits by calculating electric potentials (and currents and resistances). Pay attention to potentials, and solving circuits will become a breeze.

Voltage Source

The life mission of an ideal[1] voltage source is to maintain a potential difference (of magnitude equalling its emf value) between its two terminals. For example, a 3 V battery will keep its positive terminal 3 V higher than its negative terminal until it draws its last breath.

Resistor

An electric current I passes through a resistor R (from high to low potential) if there is a potential difference V between its two terminals. Under normal circumstances, most resistors obey Ohm’s Law.

$\displaystyle V=IR$ .

This is well understood by most students. What is often overlooked by students is that when there is no current passing through a resistor, then both ends of the resistor must be at the same potential. For example, leaving one end of a resistor unconnected results in no current passing through the resistor, and the same potential throughout the resistor.

Connecting Wires

Connecting wires are assumed to have zero resistance. A zero-ohm wire can only have zero potential difference (because the tiniest PD would have resulted in an infinite current). The entire wire is thus always at the same potential (regardless of the magnitude of the current flowing through it).

Node

Notice that all points directly connected together by zero-ohm wires must also be at the same potential. Since all these points are at the same electric potential, they can as one single point called a node. A series branch is that has only one “entry” node and one “exit” node. Parallel branches are branches that share the same two “entry” and “exit” nodes.

For example, do you think that R1, R2 and R3 are in parallel with one another?

But if you realize that the circuit has only three nodes, A, B and C,

and can be redrawn as

You should be able to see that only R2 and R3 are parallel with each other.

Example 1

What is the resistance between P and Q?

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Example 2

Evaluate the resistance between i) BC and ii) AB.

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Example 3

2016 P1 Q29

A battery of emf 24 V and negligible internal resistance is connected to a network of resistors.

What is the potential difference between junctions X and Y?

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Example 4

For both circuits, determine the range of voltage across the fixed 10 W resistor.

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Example 5

Above are two circuits for determining the unknown resistance R. An ideal ammeter has zero resistance. An ideal voltmeter has infinite resistance. For most resistance R, both circuits are equally good. However, if resistance of R is so small that it is comparable to the ammeter’s resistance, then only one circuit is good. Conversely, if R is so large that it is comparable to the voltmeter’s resistance, then the other circuit is good. Which is which?

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Example 6