Basic Electric Circuits

18.1 Resistances in Series, Parallel, and Series-Parallel CombinationsResistors in series all have the same current.

18.1 Resistances in Series, Parallel, and Series-Parallel Combinations

The sum of the voltages across each resistor equals the battery voltage; after some algebra this gives:

This formula is valid for any number of resistors in series.

18.1 Resistances in Series, Parallel, and Series-Parallel Combinations

Resistors in parallel all have the same voltage.

18.1 Resistances in Series, Parallel, and Series-Parallel Combinations

The total current is the sum of the currents through each resistor. After some algebra, we find for the inverse of the equivalent resistance:

In order to find the equivalent resistance, you will need to invert your result.

18.1 Resistances in Series, Parallel, and Series-Parallel Combinations

The equivalent resistance of resistors in series is always greater than any individual resistance in the series.

The equivalent resistance of resistors in parallel is always less than any individual resistance in the array.

18.1 Resistances in Series, Parallel, and Series-Parallel Combinations

For combination circuits, simplify piece by piece.

18.2 Multiloop Circuits and Kirchhoff’s Rules

Many circuits are not pure series-parallel combinations; more sophisticated tools are necessary to analyze them.

18.2 Multiloop Circuits and Kirchhoff’s Rules

Kirchhoff’s first rule is the junction rule:

The sum of all current entering a junction must equal the sum of all current leaving it. Giving incoming current a positive sign and outgoing a negative sign,

18.2 Multiloop Circuits and Kirchhoff’s Rules

Kirchhoff’s second rule is the loop rule:

The sum of the potential differences around a closed loop is zero.

Sign conventions for traversing batteries and resistors are at left.

18.2 Multiloop Circuits and Kirchhoff’s Rules

1. Assign directional currents to each branch of the circuit.

2. Identify enough loops so that every branch is in a loop.

3. Apply the junction rule, keeping independent equations.

4. Apply the loop rule. You should have as many equations as there are different currents.

18.3 RC Circuits

An RC circuit can be used to charge a capacitor through a resistor.

18.3 RC Circuits

When the switch in an RC circuit is closed, the current has its maximum value (all the voltage is across the resistor), and the voltage across the capacitor is zero. As charge builds up on the capacitor, the current decreases and the capacitor voltage increases.

18.3 RC Circuits

Voltage as a function of time:

Current as a function of time:

The quantity RC has the dimensions of time, and is called the time constant.

18.3 RC Circuits

A charged capacitor may be discharged through a resistor; in this case no battery is needed.

18.4 Ammeters and Voltmeters

The deflection of a galvanometer is proportional to the current.

18.4 Ammeters and Voltmeters

An ammeter measures current. In order to do this, it must be connected in series; so as not to change the existing current significantly, its resistance should be as small as possible.

18.4 Ammeters and Voltmeters

A voltmeter measures voltage. In order to do this, it must be connected in parallel across the voltage to be measured; so as not to change the existing voltage significantly, its resistance should be as large as possible.

18.4 Ammeters and VoltmetersMultirange meters have a selection of shunt and multiplier resistors, to optimize the measurement of currents and voltages of different magnitudes.

18.5 Household Circuits and Electrical Safety

Household wiring is done in parallel and protected by circuit breakers.

18.5 Household Circuits and Electrical Safety

Fuses are designed so the fuse strip melts and cuts the circuit if the current exceeds a predetermined value. Fuses are rated for different currents; the fuse rating should always match the maximum allowable current in the circuit. When a fuse burns out, it must be replaced.

18.5 Household Circuits and Electrical Safety

Circuit breakers are used in most newer homes. A bimetallic strip opens the circuit if the current becomes too high; if a circuit breaker trips, it can be reset.

18.5 Household Circuits and Electrical Safety

In order to provide needed protection, the fuse or circuit breaker needs to be on the “hot” side of the circuit.

18.5 Household Circuits and Electrical Safety

However, even on the hot side the fuse or breaker may not protect the circuit. If an internal wire touches the conductive casing of a tool or appliance, you can still get a shock. This can be

avoided by using a dedicated ground line. You can tell which devices have such a line; they have 3-prong plugs.

Review of Chapter 18

Equivalent resistance of resistors in series:

Inverse of the equivalent resistance of resistors in parallel:

Review of Chapter 18

Junction theorem: Algebraic sum of currents entering a junction is zero.

Loop theorem: Algebraic sum of voltage drops around a loop is zero.

Time constant for an RC circuit:

An ammeter measures current, and should have small resistance

A voltmeter measures voltage, and should have a large resistance