Peter's Physics Pages

An introduction to Electricity and Strength of Materials with Peter Eyland

Lecture 2 (Electric circuits)

In this lecture the following are introduced:
• Circuit Elements and Electric Circuits
• Electromotive force
• Resistors connected in series
• Resistors connected in parallel
• Series/parallel substitution of resistors in circuits
• Internal Resistance of an electric cell
• Maximum power transfer theorem

Circuit Elements and Electric Circuits

Circuit elements are devices that either produce electric currents or have potential differences formed by the current through them. An electric circuit has a number of circuit elements connected to form a closed loop or system of loops.

Electromotive force

An electromotive force (e.m.f.) is any energy source that induces an electric current.
The Voltaic cell (shown in the diagram) was the first convenient source of e.m.f. used in circuits.

The Zinc atoms in the left hand rod dissolve into the HCl leaving two electrons behind on the rod.
The Copper atoms in the right hand rod dissolve into the HCl leaving one electron behind on the rod.
Since more Zinc atoms dissolve than Copper atoms there are more electrons left on the Zinc rod than on the Copper rod.
This imbalance of electric charge creates a potential difference between the rods with the Copper rod being effectively more positive.
Since electric current is a flow of positive charge, conventional current will flow out from the positive terminal (Copper), around an external circuit, and back in to the negative terminal (Zinc).
The emf appears as an electric potential difference between the terminals of the cell.

 The circuit symbol for such an electric cell (a battery is a collection of cells), which provides a current, is a long line and a short line at right angles to the connecting wires. The long line is the positive side and the short line is the negative side. Normally current comes out the positive side.

Resistors

 Resistors are devices that resist the free flow of electric current through them and produce a potential difference across them when a current flows. They used to be solid cylinders of a Carbon based material. Recently they became ceramic cylinders with a coating of material on the outside. A spiral groove is cut to create a spiral strip around the outside of the cylinder. Note that potential difference is across an element and current is through an element.

 There are two symbols and notations used for resistors in circuits. Examples of Notation: American: 2.7Ω and 3.3kΩ International: 2R7 and 3k3 There is no unit written because the circuit symbol (a rectangle for resistors) indicates the type of device and the unit

Resistors are labelled by using a colour coding of bands around the resistor. The following table gives the coding.

Resistors connected in series.

 Resistors are connected in series when: (i) the same current flows through them (i.e. from one to the next without branching), and, (ii) they are connected + to - (the low potential side of the first is connected to the high potential side of the next to form potential steps down).

Example C1

 Three resistors are connected in series with each other and an e.m.f. as shown in the diagram. Find the equivalent resistor for the three resistors (i.e. one resistor to replace the three and produce the same effect).

When resistors are connected in series, a common current, I, flows through them.
In a series circuit, since the current is common, the approach is to add up the potential differences across each resistor. The is gives the following:

For resistors connected in series the equivalent resistor is a replacement resistor whose resistance is the sum of the resistances.

Resistors connected in parallel.

 Resistors are connected in parallel when they (i) have the same potential difference across them, and (ii) are connected + to + and - to -

Example C2

 Three resistors are connected in parallel with each other and an e.m.f. as shown in the diagram. Find the equivalent resistor for the three resistors (i.e. one resistor to replace the three and produce the same effect).

When resistors are connected in parallel, a common potential difference, V, acts across them.
In a parallel circuit, since the potential difference is common, the approach is to add up the currents through each resistor. This gives the following:
 For two resistors in parallel it is convenient to note that

Series/parallel substitution of resistors in circuits

A circuit, which consists of an electric cell and a number of resistors connected in series and parallel, can be analysed by successive substitution of equivalent resistors.

For each resistor this will eventually give
• the current through it, and
• the potential difference across it, and
• the power dissipated by it.

Example C3

 For the circuit shown in the diagram, find the current through; the potential across; and the power dissipated in each resistor.

The resistors are connected in series because the same current flows through each.
The three resistors can be substituted by a single resistor, as follows:

The common current in the circuit is:

The potental differences are:

These add up to 240V so the answers are consistent.

The powers dissipated are:

since

These are consistent.

Example C4

 For the circuit shown in the diagram, find the current through; the potential across; and the power dissipated in each resistor.

The common potential difference is 240V.

The powers dissipated are

The total power dissipated from this by addition is 5952 W.
The total power is also P = VItotal = 240 × 24.8 = 5952 W.
These are consistent.

Internal Resistance of an electric cell

In an electric cell (or battery of cells), the conversion of chemical energy to electrical energy is not 100% efficient. There is some waste heat produced. This waste heat is accounted for by modelling it as heat dissipated from an internal resistance, r. A real electric cell (or battery of cells) can then be modelled as an e.m.f. plus an internal resistance.

 When current is drawn from the cell, the product of the current and the potential difference across the modelled internal resistance accounts for the internal energy loss. The potential difference, V, which appears across the terminals, will be less than the stated e.m.f., E, because of the potential difference across the internal resistance. V = E - rI

Example C5
An electric cell is connected to a 4Ω resistor and a current of 3A flows through the resistor. When the same electric cell is connected to a 10Ω resistor a current of 2A flows. Find the emf and internal resistance of the cell.

Answer C5
 In the 1st circuit: In the 2nd circuit:

Since E is the same

From the first circuit equation, the emf is given by

Thus the emf is 36V and the internal resistance is 8Ω

Example C6
An electric cell has an emf of 12.0 V and a potential difference of 11.5 V appears across the terminals when a current of 0.25 A is drawn from it. Find the internal resistance and the heat dissipated by the internal resistance.

Answer C6

Consistency check: The total power supplied is E·I = 12 × 0.25 = 3W
The power delivered to the circuit is V·I = 11.5× 0.25 = 2.875W
The power lost in the energy change (i.e. due to the internal resistance) is 3 - 2.875 = 0.125W, as above.

Maximum power transfer theorem

The maximum power that can be delivered by an electrical source depends on the internal resistance of the source.

 The circuit diagram is: The circuit potential differences are related by: Power dissipated in the external load is given by:

For the maximum power delivered to the external circuit, we need find the right resistance by setting

This goes to zero when the numerator is zero, i.e. when when R = r

When the external load equals the internal resistance then maximum power is transferred from the source

Summarising:

Circuit elements are devices that produce currents or have potential differences formed across them.

An electric circuit has a number of circuit elements connected to form a closed loop or system of loops.

An Electromotive force (e.m.f.) is any energy source that induces an electric current.

Resistors are connected in series when the same current flows through them and they are connected + to -.

Approach a series circuit by adding up the potential differences around the circuit.

For resistors in series:

Resistors are connected in parallel when they have the same potential difference across them and they are connected like to like.

Approach a parallel circuit by adding up the currents through the circuit.

For resistors in parallel:

For two resistors in parallel:

The waste heat of an electric cell is accounted for by thinking of it as heat dissipated from an internal resistance.

When the external load equals the internal resistance of an e.m.f. then maximum power is transferred from the source.

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