A: Before I dive into the relationship between resistance and voltage in a filament lamp, it is important to first understand what type of a conductor a filament lamp is. We say that it is non-ohmic - meaning it is a conductor that does not show a linear relationship between potential difference and current. Immediately we realise that filament lamps therefore do not have constant resistance - it will vary depending on the voltage.
Now, in order to understand this relationship, it helps to look at the filament lamp’s I vs. V graph:
The gradient will give us value for the inverse of resistance - notice how the claim of “varying resistance” is justified with how it’s gradient keeps on changing with voltage!
We can see how as we go along increasing values of V, the gradient of the line keeps on decreasing, leading to an increase in resistance. Think of it this way: as you walk along the x axis (showing voltage V), you notice that the line gets less steep (meaning the gradient is getting smaller). And since m = 1R, decreasing gradient means resistance is increasing!
So we have established a general relationship: for a filament lamp, as voltage increases, resistance increases as well. But we would like to also understand why this is the case. For that, we have to look at the filament inside, made of tungsten. This metal naturally has a high resistance; so when more voltage is applied, more energy is used per unit charge in passing through the filament. This ultimately creates a strong heating effect, causing the filament inside to glow. But inadvertently, this increasing temperature also causes the ions inside the filament to vibrate more, thereby making the flow of current even more difficult. Hence, resistance increases.