Either this question is poorly phrased, or there is additional information related to it.
We know that increasing the voltage will increase the current through both the conductor and the semiconductor, therefore the question here is: which of the currents will increase more?
We know that the conductor's current increases linearly with voltage, but what behavior will semiconductor show?
This is the point where we can't say anything without additional information:
1) Are metal-semiconductor contacts ohmic or rectifying?
2) Is semiconductor intrinsic or extrinsic (is it doped)?
3) Should we consider second order effects (like conductivity change due to heating, mobility degradation under heavy bias, etc.)?
Assuming that there is no additional information provided with the question, I would guess that the contacts are ohmic and their resistance may be neglected, the semiconductor bulk is intrinsic and we should not consider second order effects. In this case, since the current is equal for the same bias, the resistances of semiconductor and conductor are equal, and the current-voltage characteristics will be the same, and the correct answer will be C.
How can a charge neutral substance have a potential?
To say that a PN junction has built-in potential isn't to say that the PN junction has a potential relative to ground or infinity etc.
How can there be an inherent potential in a doped semi-conductor if it
is charge neutral?
Charge has been separated within the PN junction and, thus, there is an electric field across the depletion region and an associated potential difference. A charged capacitor is neutral but there is a potential difference (voltage) across the dielectric. There are some similarities but...
However, I don't understand why we can't measure the voltage drop
across the PN block with a voltmeter.
As explained, for example, here, the built-in potential is not readily measured with, e.g., a voltmeter. In other words, this question has been asked here several times (which means you are not the only one perplexed by this - most are at first) and there are good answers already available.
Best Answer
That's not how I learned it. It has to do with energy gaps between valence band where the electrons at the highest energy are, and the conduction band, where an electron can come free of its atom. In metals these bands overlap and electrons move free within the metal's lattice, and that's what gives metals their typical shine.
Pure semiconductors are isolators at cryo-temperatures. But doping them with N-type material will cause doping atoms to bond with the semiconductor, and then there's one superfluous electron which, like in metals, can move freely through the lattice, and thus conduct electricity.
(from this excellent site)