Direct Current Circuits

Resistivity

Resistance depends on three things inside a component:

What is resistivity?

Resistivity is defined as the resistance of a 1m length with a $1m^{2}$ cross sectional area. It is measured in ohm-metres ($\Omega m$).

Resistivity can be calculated by the following equation:

$$ \rho = \frac {RA}{L} $$

Where $\rho$ is resistivity, $R$ is resistance, $A$ is cross-sectional area and $L$ is the length.

Worked Example

The heating element in a toaster has a radius of 0.2mm, a length of 0.8m and a resistivity of $1.10 \times 10^{-6}$.

$$ \rho = \frac {RA}{L} \\
\eqalign{
1.10 \times 10^{-6} &= \frac {R \times (\pi(0.2 \times 10^{-3})^2)}{0.8} \\
1.10 \times 10^{-6} \frac{0.8}{1.26 \times 10^{-7}} &= R \\
R &= 7\Omega
}
$$

Direct Current Circuit Rules

Cells in Series and Parallel

Kirchoff's Laws

Direct current circuits follow Kirchoff's Laws which explain the current and emf behaviour.

Kirchoff's First Law

This states that the current entering a junction is equal to the current leaving it. In a circuit, the charge isn't used up or lost meaning whatever flows into a junction will leave it. In a series circuit, this causes the current to be equal everywhere in the circuit.

In a parallel circuit, the current depends on the resistance of each loop. Regardless of how the current is divided, the current entering the junction will always equal the current leaving it.

Ijunction

Kirchoff's Second Law

Kirchoff's Second Law states that the total emf of a series circuit is equal to the sum of the potential differences across all the component.

Vparallel

Resistance Laws

These laws can be used to determine the resistance of components in circuit.

EMF and Internal Resistance

Resistance is caused by electrons colliding with atoms of the material and losing energy. In batteries, chemical energy is used to make electrons move. As they move, they collide with the battery, creating an internal resistance.
emf_ir

With no internal resistance the emf would equal the terminal pd, though as there is an internal resistance energy is lost. This energy are the lost volts. The following equation can be used to calculate emf:

$$ \epsilon = IR + Ir $$

Uses of Internal Resistance

In order to deliver a high current, the source must have a low internal resistance. High voltage supplies are designed with a high internal resistance so that if a short circuit occurs, only a small current flows which is safer.

Potential Divider

A potential divider is a circuit with a voltage source and some components in series. The potential of the supply is divided between the components so if a $2\Omega$ and a $3\Omega$ resistor are in series, the $2\Omega$ gets 2/5 of the voltage and the $3\Omega$ gets 3/5 of the voltage. The equation for a potential divider is:

$$ V_{out} = \frac {R_{1}}{R_{1} + R_{2}}V_{s} $$
potential_divider

LDR and Thermistors

LDRs and thermistors are often used in conjunction with a transistor in a potential divider circuit.

Potentiometers use the potential divider principle as a slider which moves the relative sizes of $R_{1}$ and $R_{2}$, changing the voltage across a component.

potentiometer

Alternating Current

An alternating current is a current that changes with time. In an alternating circuit, the voltage rises to a peak before decreasing to a trough, forming a sinusoidal pattern. To measure the voltage over time, an oscilloscope is used.

ac_wave

By measuring the distance between successive peaks, the time period can be determined, which can be used to calculate the frequency.

$$ f = \frac {1}{T} $$

Root Mean Square

An alternating current with a peak voltage of 2V will actually be below 2V most of the time. This means the power output compared to a direct current supply will be much less. In order to compare the voltages, the root mean square average can be calculated:

$$ V_{rms} = \frac {V_{0}}{\sqrt{2}} \\
I_{rms} = \frac {I_{0}}{\sqrt{2}} $$

© Andrew Deniszczyc, 2018