Revise Practical circuits for Cambridge 9702 (10.1)

Every power source, like a battery or power pack, has an electromotive force (e.m.f., ε). This is the total electrical energy it supplies per unit charge to drive current around a complete circuit. It’s like the 'push' for the charges, representing the maximum potential difference the source can provide when no current is drawn (on an open circuit).

$\varepsilon = \frac{W}{Q}$ (where W is energy/work done, Q is charge)

No power source is perfect. Inside, it has a small but significant internal resistance (r) due to the materials it's made from. This resistance causes some of the energy to be wasted as heat when current flows, creating a 'lost voltage' internally. This means the voltage available to your external circuit, called the terminal potential difference (V), is always less than the full e.m.f. when current is flowing.

$\varepsilon = V + Ir$ OR $\varepsilon = I(R + r)$ (where V is terminal p.d., I is current, R is external resistance)

An important consequence of internal resistance is its effect on the power delivered to the external circuit. The power delivered to the external load resistor R is given by $P = I^2R$. Since $I = \varepsilon / (R+r)$, the power is $P = (\frac{\varepsilon}{R+r})^2 R$. A key result from this relationship is the maximum power transfer theorem. This states that the maximum power is delivered to the external load when the load resistance R is equal to the internal resistance r of the source.

Condition for maximum power transfer: $R = r$

When this condition is met, the power delivered to the load is $P_{max} = \frac{\varepsilon^2}{4r}$. However, note that under these conditions, an equal amount of power is dissipated as heat inside the source, making the process only 50% efficient.

To analyse any complex circuit, we rely on Kirchhoff's Laws. His First Law, also known as the junction rule, is all about the conservation of charge. It simply states that charge cannot accumulate at a junction; the total current flowing into a junction must equal the total current flowing out.

$\sum I_{in} = \sum I_{out}$

Kirchhoff's Second Law, the loop rule, reflects the conservation of energy. It tells us that for any closed loop in a circuit, the algebraic sum of the e.m.f.s is equal to the algebraic sum of the potential differences (p.d.s) around that loop. A common sign convention is: e.m.f.s are positive if they drive current in the loop's direction, and p.d.s across resistors are negative as they represent an energy loss.

$\sum \varepsilon = \sum IR$

When components are connected in series, they form a single path for the current. This means the current is the same through every component. The total voltage from the source is divided among them in proportion to their resistance ($V_1/R_1 = V_2/R_2$), and the total resistance is the sum of individual resistances.

$R_{total} = R_1 + R_2 + ...$

In a parallel circuit, components are connected across the same two points, creating multiple paths for the current. Here, the potential difference (voltage) across each parallel branch is identical. The total current from the source splits, with each branch drawing current inversely proportional to its resistance. The total resistance is always less than the smallest individual resistance.

$\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$

A potential divider is a circuit using two or more series resistors to provide an output voltage ($V_{out}$) that is a precise fraction of the input supply voltage ($V_{in}$). By choosing the right resistance values, you can 'tap off' exactly the voltage you need for a specific part of your circuit. It's a very common and useful circuit configuration.

$V_{out} = V_{in} \times \frac{R_{out}}{R_{total}}$ (where $R_{out}$ is the resistance across which $V_{out}$ is measured)

You can make a potential divider variable using a potentiometer (a three-terminal resistor with a sliding contact) or a sensor like a thermistor (resistance depends on temperature) or Light Dependent Resistor (LDR) (resistance depends on light intensity). As the sensor's resistance changes, the output voltage of the divider changes, allowing it to act as a responsive sensing circuit for applications like thermostats or automatic lighting.