4.3.3 Potential dividers

Potential divider circuits can vary the p.d. across an output (e.g a lamp) when connected to a fixed point. These are useful in circumstances such as having a 10V battery but only needing 6V for the task you are about to perform.

Basically, there are two resistors in a series circuit. All you have to do is connect a circuit across one of these resistors. The p.d. into this circuit (which is Vout of the potential divider circuit) can be varied from zero to maximum depending on the resistances of R1 and R2.

This  means that the p.d. across each resistor in a potential divider depends on the resistances of the individual resistors...

V1/V2 = R1/R2

We can determine Vout by the potential divider equation:

Vout = (R2/(R1+R2)) x Vin

You can also 'load' a potential divider circuit. This refers to adding an additional resistor/component to the Vout part in parallel which overall decreases the total resistance of this part which lowers Vout. A small additional resistance in parallel significantly reduces Vout.

The potentiometer
To vary Vout we can replace one of the fixed resistors with a variable resistor. A potentiometer is a type of variable resistor that has 3 terminals and a sliding contact. If you adjust the contact the p.d. between two of the terminals will vary. Potentiometers are very compact which is useful so they can be used for portable electronic devices etc.

Temperature sensing circuits
To vary Vout we can replace one of the fixed resistors with a variable resistor.

Replacing the variable resistor with a fixed resistor allows Vout to vary depending on the temperature of the surroundings (as temperature increases resistance of the thermistor decreases as thermistors have a negative temperature coefficient).

Light sensing circuits
A similar principle to temperature sensing circuits only replace the variable resistor with an LDR. As light intensity increases, resistance falls.

4.3.2 Internal resistance

It is important to understand that different power sources have different internal resistances. E.g if a large current is needed, a power supply with a small internal resistance is used. If a small current is needed, a power supply with a large internal resistance is used.

So, what is internal resistance? Basically, when there is a current in a power source work has to be done by the charges to move through the power source. The p.d. measured at the terminals of the power source (the terminal p.d.) is always less than the actual emf (provided there is a current in the circuit) - the difference is the lost volts.

From K2 we can determine this equation:

electromotive force (emf) = p.d. + lost volts

It is important to realise that changing the current will change the lost volts and consequently the terminal p.d.. An increase in the current means that more charges travel through the cell each second meaning overall more work is done by the charges. This increases the lost volts which subsequently reduces the terminal p.d.

We can apply V=IR to lost volts too (lost volts = Ir, where r is the internal resistance). This means that total emf is equal to IR + Ir...

emf = I(R+r)

We also need to be able to determine the internal resistance of a cell/source of emf:

• Create a series circuit with a variable resistor
• Introduce a voltmeter across a power supply
• Have an ammeter in the circuit (it doesn't matter where as it is a series circuit)
• Change the resistance of the circuit - this will draw different currents from the power source.
• Rearrange E= V+Ir to give V = -Ir + E
• Plot a graph of V against I
• The gradient will be -r, the y intercept will be the emf

4.3.1 series and parallel circuits

Kirchoff's Laws
Kirchoff's first law states that the sum of currents into a point always equals the sum of currents out of a point. This is conservation of charge.
Kirchoff's second law states that the sum of the emfs is equal to the sum of the pds around a closed loop. This is conservation of energy.

During GCSE you will have come across series and parallel circuits (I hope!). Here is a reminder...

A series circuit has only one path for current. From K1 we can deduce that, since charge is not used up, the rate of flow of charge is the same at all points in the circuit. Since current is the rate of flow of charge this means current is the same at all points in the circuit. From K2 we know that the emf is shared between each component as there is only one closed loop. Adding additional components in series reduces the emf that is shared between the original components. We know that V=IR and I is constant (in series circuits). The total V = V1+V2... this means that the total resistance (R)= R1+R2...

A parallel circuit has more than one possible path for current. The resistance of each path determines how much current will flow down it. The greater the resistance, the lower the current that passes down it (V=IR and V is constant). Each branch of a parallel circuit forms it's own loop. from K2 we can deduce that the pd around each individual loop is equal to the emf from the power supply. In parallel circuits, V is constant. This means that I/V=I1/V+I2/V..... I/V = 1/R. This means that 1/R = 1/R1+1/R2...

These rules can be used to create a circuit with a required resistance.


When using more than one emf source, we must look at which polarity the emfs are in. We need to think about the internal resistance when combining emf sources in series and parallel.

In series: the emf will change but resistance will be constant

In parallel: the emf will be constant but resistance will change

Image source: Kerboodle OCR A Physics textbook p183

4.2.5 Power

Electric circuits are used to transfer energy from one place to another. Whenever there is a current in a circuit energy is transferred from the power source to the component. Opening a switch/turning off the power source will result in no current as no energy is transferred.

Power is the rate of energy transfer between each electrical component. It is measured in watts and can be calculated using the equation:

P=IV

Using V=IR we can get two more equations for power...

P= I²R and P=V²/R

The derivation:

P=W/t

W=VQ

P=(VQ)/t

I=Q/t

P=IV

If we substitute P=W/t (--> W=Pt) into P=IV we get W=VIt. This is an equation for energy transferred.

Last bit in this section (yayy). We just need to know about the kilowatt hour and how to calculate the cost of energy...

The energy transferred to a device depends on how long the device is used for and the power of the device.

One joule is a very minuscule amount so instead electricity bill people use the kilowatt hour (kWh). This is the energy transferred by a device with a power of 1kW operating for a time of 1 hour. It is equal to 3.6MJ.

4.2.4 Resistivity

Four different things affect the overall resistance of a wire: the material, the length, the cross-sectional area, and the temperature. resistance is directly proportional to length and inversely proportional to cross sectional area.

Resistivity can be used to describe the electrical property of a material (resistance only describes the electrical property of a single component). E.g copper wires may have a different resistance (e.g if they are different lengths) but overall copper has a unique resistivity. We can use the above facts (that resistance is directly proportional to length and inversely proportional to cross sectional area) to make an equation with resistivity as the constant:

R = (ρL)/A

Resistivity has the symbol pho (ρ) and the unit ohm meter (Ωm). Resistivity is defined as the product of the resistance of a component made of the material and it's cross sectional area, divided by it's length. 

A bit of a mouthful if you ask me.

Like resistance, resistivity also varies with temperature.

How would we determine the resistivity of a material?...
Investigate how the resistance of a wire varies with length. Measure values of potential difference across varying lengths of wire. Use V=IR to determine R. A graph of R against L will give a gradient of ρ/A. Multiply the gradient by A and you have ρ, the resistivity.

Good conductors have a very small resistivity whilst insulators have a very large resistivity.

In this section we also need to know about NTCs and the variance of resistance with temperature of a thermistor. I have put this information in this section.

4.2.3: I-V characteristics

I-V characteristics show the relationship between the electric current in a component and the potential difference across it. This information is commonly collected using a potentiometer/variable resistor. One main point to look at is whether the component behaves the same way if the current is reversed (goes through it in the opposite direction). We need to know the shape of I-V graphs for a few components...

Resistors

• Fixed resistors are designed to keep constant resistance regardless of temperature fluctuations
• The potential difference across a resistance is proportional to current so the resistor can be said to obey Ohm's law.
• This means it is an ohmic conductor (just a conductor that obeys Ohm's law)
• The resistors behave in the same way regardless of polarity
• The shallower the gradient the greater the resistance

Filament lamps

• The potential difference across a filament lamp is not proportional to the current through the resistor so a filament lamp does not obey Ohm's law (it is a non-ohmic conductor)
• The resistance of a filament lamp is not constant, it increases as the p.d. increases (remember: shallower gradient = greater resistance). The increase in resistance is caused by the wire getting hot as it glows (this point here).
• The filament lamp behaves the same way regardless of polarity

Light emitting diodes

• Diodes only allow current in one direction - its behaviour depends on the polarity. With a negative p.d there is no current - resistance is infinite and the diode will not conduct. As p.d. increases resistance gradually starts to drop. This is known as the threshold p.d.. Above threshold p.d., an increase in p.d. leads to a larger decrease in resistance and the diode begins to have very little resistance. The value for threshold p.d. varies depending on the colour (wavelength) of light they emit.
• The potential difference across a diode is not directly proportional to the current through it therefore it does not obey Ohm's law (it is a non-ohmic component)
• Resistance is not constant (seen by a fluctuating gradient)

Thermistor

Thermistors are temperature sensing components with a negative temperature coefficient. This means that their resistance DROPS when temperature increases (this is the opposite of a metal wire). This effect can be explained in terms of number density. In certain semiconductors (the ones with negative temperature coefficients) the number density of the charge carriers increases as the temperature increases. Often the resistance drop is large meaning a small change in temperature can be detected by monitoring the resistance of the thermistor. Thermistors are used in thermostats (controlling heating/air-conditioning units), to measure the temperature of engines and electrical devices to ensure they do not overheat, and in simple thermometers.

Like LEDs and filament lamps, thermometers are non-ohmic. With a thermistor, as the current increases temperature also increases (like a filament lamp) but this leads to a DROP in resistance because the number density of the charge carriers increases. The graph is pretty similar to a filament lamp (don't get them confused!) but the gradient increases for a thermistor (= decreased resistance) and the gradient decreases for a filament lamp (= increased resistance)...

We need to know how to investigate the relationship between resistance and temperature. This can be done using an ohmmeter and a water bath, or also an ammeter and voltmeter at different temperatures then use V=IR to calculate resistance. Results for this experiment will vary slightly with different thermistors. This can ensure that the best thermistor is selected for the right application.

LDR

Light dependant resistors are little electrical components which change their resistance depending on the light intensity. Some uses are in sports/street lamps/brightness metres in phones and laptops. An LDR is made from a semiconductor in which the number density of charge carriers changes depending on the incident light intensity. Dark conditions = low number density = high resistance, light conditions = high number density = low resistance.

We need to know about the relationship between the resistance and light intensity or an LDR and how to experiment it...

Image credit: Kerboodle OCR A Physics textbook p160

Vary the distance from the LDR to the constant light source (the lamp). This will change the intensity of light recieved by the LDR. Putting a small black cardboard tube around the LRD will redice background light interfering. The results will give a calibration curve that you can use to read unknown readings off...

Image credit: Kerboodle OCR A Physics textbook p160

Image sources: Kerboodle OCR A Physics textbook p.149-159

4.2.3: Resistance

All components in a circuit have their own resistance. Resistance is the ratio between the pd across a component and the current in a component. The unit of resistance is the ohm. The ohm is the resistance of a component when a pd of 1 V is produced per ampere of current. In other words, 1 Ω is 1VA⁻¹.

Resistance can be calculated using the equation V=IR.

Ohms law states that or a metallic conductor kept at constant temperature, the current in the wire is directly proportional to the p.d. across its ends.

It is important to remember that temperature affects resistance (which is why Ohms law states at a constant temperature). In the example graph below, the p.d of the wire remains constant (1.5V) but the current deceases with time. This is because resistance increases with time (by looking at V=IR we can see that if V is constant then I is inversely proportional to R). Over time, the temperature of the wire increases and as the wire gets hotter its resistance increases. This is because the positive ions inside the wire have more internal energy therefore they vibrate with greater amplitude about their mean positions. The frequency of collisions between charge carriers and positive ions increases and this means that charge carriers do more work (transfer more energy as they travel throughout the wire).

Photo credit: kerboodle OCR A physics textbook p.147

As well as temperature, the material of the wire, the length of the wire, and the cross-sectional area of the wire all affect resistance.