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### Further Maths

• A-Level Physics

# Energy, Power & Resistance

Updated: Aug 22

### Potential Difference

Potential Difference (p.d.) is the work done per unit charge – the amount of energy converted from electrical to another form per coulomb of charge passing through a component.

W = VQ work done = p.d. × charge

### Electromotive Force

Electromotive Force is the amount of energy converted into electrical energy per coulomb of charge through the source:

W= εQ work done = e.m.f x charge

Essentially, p.d. is voltage used up in a circuit, whereas e.m.f. is the voltage provided to the circuit by a cell or power supply.

Both p.d. and e.m.f. are often referred to as voltage, because they both shave the Volt as their unit.

The Volt is 1J of energy per 1C of charge.

## Accelerating Electrons

A cathode-ray tube (also called an electron gun) is used to accelerate electrons. A hot metal filament releases electrons by thermionic emission, and then these are accelerated in a vacuum between the filament and a metal plate with a small hole in it. The two are connected and a very high p.d. is applied, making the filament a cathode and the plate an anode. The electrons released from the filament are attracted to the positively charged anode plate, and pass through the hole. This creates a concentrated beam of high velocity electrons.

When an electron is accelerated in this way, the energy transferred equals the wok done on the electron, which equals the kinetic energy:

eV = 1/2 m v^2 work done on electron = gain in kinetic energy of electron

## Resistance

All conductors have resistance that obstructs the flow of charge through them. The higher the resistance of a conductor, the more energy is required to push the charge through, as more energy is lost in the process.

### Ohm's Law

Ohm's Law states that the current flowing through a conductor is directly proportional to the p.d. across it, provided that the physical conditions (such as temperature) remain constant.

V = IR p.d. = current x resistance

The units of resistance are Ohms, Ω, defined as the resistance of a conductor when 1 volt produces a current of 1 ampere through that conductor.

### I-V Characteristics

To investigate the I-V relationships of components, a test circuit as shown can be set up and the current recorded for a variety of potential differences. Taking averages increases reliability, measuring to 3 sig-figs increases accuracy. A graph can be plotted to see its ohmic or non-ohmic properties.

The ammeter must be connected in series, and the voltmeter in parallel.

A Resistor/Metallic Conductor obeys Ohm's Law - the I-V characteristic is a straight line through the origin, where the inverse of the gradient is the resistance (R = V/I).

A Filament Lamp is an example of temperature affecting resistance. As the p.d. increases, so does the brightness and temperature of the bulb and with it, its resistance.

This is because at higher temperatures, the positive ions in the filament have greater internal energy, meaning they vibrate more. Therefore, the probability of the charge carriers colliding with them is greater. In this instance, Ohm's law does not apply.

A Diode only works with current flowing thorough it in one direction - this is known as the forward bias. Typically, there is a threshold of around 0.6V before they conduct at all.

Light Dependent Resistors and NTC Thermistors (Negative Temperature Coefficient) lose resistance as the energy on them increases. This is because they are semiconductors, that release more electrons for conducting when they have more energy. This means the electron density increases, so the current increases and resistance decreases.

## Resistivity

Resistivity is a property of a material, as different structures make for different conducting properties. It is defined as the resistance from one side to another of a 1m cube.

R = ρL/A resistance = resistivity × length / cross-sectional area

This relationship means that resistance is directly proportional to length, while it is inversely proportional to the cross-sectional area:

R ∝ L R ∝ 1/A

Resistivity is measured in ohm meters, Ωm

### Investigating Resistivity

To determine resistivity, you need to work out the cross-sectional area using a digital caliper. Then, clamp the wire in place with a voltmeter in parallel and an ammeter in series with a fixed power supply. Vary the length of wire (use a ruler) and measure voltage and current to establish the resistance. Repeat at several different lengths.

The resistivity of a metal increases with temperature. This is because charge is carried by moving electrons, but heat makes this motion harder as the ions in the structure vibrate more (see I-V characteristics of a filament lamp above). The resistivity of a semiconductor decreases with temperature, as more electrons are released for conducting (See LDR/Thermistors above).

## Energy & Power

Power is the rate of transfer of energy, where one Watt is one joule of work done per second.

P = IV power = current x voltage

Using the equation V = IR, we can rewrite this power equation to other, often more useful forms:

If power is energy per second, then we can use it to work out the electrical energy (work done):

W = Pt work done = power x time

Again, this can be rewritten in multiple ways:

A joule is a very small unit of measuring energy. In fact, it is so small that it is rarely used commercially, because we typically use thousands or millions of joules at a time. Therefore, the Kilowatt-hour is used domestically:

1 kWh = 3.6 million joules

This can quickly be calculated:

1 kW = 1000W

1 h = 60x60s = 3600s

1 kWh = 1000W x 3600s = 3,600,000 Ws = 3,600,00 J

Energy Transferred (kWh) = Power (kW) x Time (h)

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