RL = (VL/I) × 1000 Ω
RS = 1000 × (VS - VL)/I Ω
Total Resistance = R = (RL + RS) Ω
Load line for the series circuit : VS = VL + I × RS -> VL = -IRS + VS
The voltage drop across the lamp can be computed graphically by drawing the load line for the circuit and the V-I curve for the lamp in the same graph and taking the intersection point voltage.
When I = 50 mA -> (a)RL = 88 Ω (b)RS = 60 Ω (c)RL = 100 Ω, RS = 64 Ω, R = 164 Ω
When I = 150 mA -> (a)RL = 480 Ω (b)RS = 64 Ω (c)RL = 467.33 Ω, RS = 66 Ω, R = 533.33 Ω
Percentage increase in resistance (I= 50 mA to I = 150 mA)
(a) RL = 100 × (480-88) / 88 = 445.45 %
(b) RS = 100 × (64-60) / 60 = 6.67 %
(c) RL = 100 × (467.33-100)/100 = 367.33 %
RS = 100 × (66-64) / 64 = 3.125 %
R = 100 × (533.33-164) / 164 = 225.2 %
RESULTS
VL Comparison
Percentage Increase I = 50 ->150 mA
DISCUSSION
A definition to the electrical resistance is given by the Ohm’s law. Ohm's law is an assertion that the current through a device is always proportional to the potential difference applied to the device.
i.e. R α V / I -> R = V / I (with proper units) -> V = I R
The assertion is correct only in certain situations but still, for historic reasons, the term ``law'' is used. The proportionality constant is called the resistance of the device and is related to the resistivity of the material from which the device is made. Since resistivity varies with temperature, so too does the resistance. As the device heats up, there is increased thermal activity at the atomic level and the material's ability to provide charge carriers changes.
The resistance of a resistor is given by the equation
R = R0 (1+ α θ)
where R - the resistance at temperature t, R0 - the resistance at temperature t0 (standard), α – the temperature coefficient and θ = (t- t0). But this equation is only valid if the temperature coefficient doesn’t vary with temperature. That is the R = f (t) function of a resistor must be linear. Also it is said that if α >0 it has a positive temperature coefficient and if α <0 it has a negative temperature coefficient.
Standard resistors show very small variations and their temperature coefficients typically -> 0. So for most applications they are considered constant i.e. R ≈ k. But in the case of a filament lamp there is very much difference. Its resistance is due to the thin metal "filament" inside the lamp typically made out of tungsten – a material known to have a high positive temperature coefficient. This results in a large variation in resistance over temperature changes. Here the temperature change doesn’t necessarily mean it’s the environmental temperature. The interesting thing happen is when current flows across such a resistor the Ohmic enrgy lost (E = I2 R or V2 / R) dissipated as heat increases the temperature of the filament. This in turn increases the resistance.
This vast resistance variation due to the heavy heating of the high temperature coefficient material has both advantages as well as disadvantages. When we consider the lighting effect of the bulb, light is produced by the heavily heated tungsten material emitting electro-magnetic waves. So we call the human visible portion of this wave band as light. This means a practical filament should be heated up to a high temperature in few mili-seconds. So we find it extremely useful, the large increase in the resistance of the filament when current flowing through it.
What happens is when we switch on the lamp the current increases from its initial value zero, flowing through the filament whose initial resistance is cold resistance. This produces heat and increases the resistance. Because of increased resistance and of the still increasing current it produces more heat. This process is repeatedly occurred until the current, resistance and temperature comes to a balance condition in few mili-seconds. At this time the bulb emits visible light due to heavy heating. Without the high resistance variation of the filament this process might take quite time which is not practical. Another advantage is that we get our current limited with the increased resistance after the balanced condition.
There are also some disadvantages of this resistance variation. One thing is the loss of energy due to the heating with the increased resistance. Only part of the energy supplied is emitted as visible light, so there’s a high inefficiency.
When the current changes from I = 50 mA to 150 mA the standard resistor had a 6.67 % increase in resistance and the filament resistance a 445.45 % and the series combination a 225.2 %. This clearly confirms the theory discussed above.
We got some practical errors in this experiment. The cold resistance 58.0 Ω was not the one obtained from calculation at initial current. This maybe due to the voltmeter finite resistance (ideally this should be ∞ Ω) and the ammter internal resistance (ideally this should be 0 Ω) and also the resistances associated with connecting wires, etc.
The standard resistor we used was a 100 Ω / 2 A rheostat adjusted to 58 Ω. Here the rated current 2 A is very much greater than the maximum current we allowed to flow in the experiment which is 160 mA. The meaning of the rated current is the maximum current the resistor can follow up to, without considerable heat generation i.e. maintaining resistance nearly a constant. So if we used a 150 mA rated rheostat which is even below the 160 mA, that resistor might not behave as a standard resistor. Instead it will show a higher degree of increasing resistance than before due to the considerable heating effect. Of course this might not be large as the filament resistance.