A Lightning rod is fixed to the building and it is not given earthing then what happens? How does the lightning reach the ground through the building?
[Physics] What happens when a lightning rod is not earthed
electricitylightning
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The lightning rod is based on two principles theorized by Benjamin Franklin. Lightning dissipation theory, and lightning diversion theory.
Lightning Dissipation Theory
This theory says that if you point a pointy metal object toward a polarized cloud, the metal object will be able to bleed off some of the energy from the cloud. Thus preventing a lightning strike.
This theory can actually be demonstrated, using a Van de Graaff generator and a nail. This YouTube video demonstrates the theory.
While this theory holds up on the small scale, it's been shown not to be effective at dissipating the large amount of energy built up in a storm. Fortunately, the design of the dissipation device (lightning rod) is also a great diversion device.
Lightning Diversion Theory
The lightning diversion theory says that if you provide a preferable path for the energy to travel along, there's a high probability the energy will follow that path.
Lightning rods are designed to be the highest objects around. This puts them closer to the polarized cloud, and reduces the distance the lightning must travel through the air. They are also made from conductive materials, and are connected to the earth through highly conductive materials. This provides a low resistance path to ground, making it a preferable path for lightning to follow.
While both theories hold up in the laboratory, only diversion theory seems to offer a viable lightning protection system.
Yes, it's possible. Though, I'm sure the engineers of the Sears Tower took that into account.
Catastrophic failure of the rod is pretty straightforward. Lighting is a ton of electrical current, and the rod has some resistance. Current through a resistance makes heat by Joule heating, which says that the power is proportional to the resistance times the square of current.
A big strike can have currents up to 120 kA. The resistivity of copper is about $1.68 \cdot 10^{−8} \Omega \mathrm m$, so the resistance of a typical ground rod used in US residential construction (5/8 in in diameter, 8 feet long) is something like:
$$ \frac{1.68 \cdot 10^{−8} \Omega \mathrm m \cdot 2.5 \mathrm m}{2\cdot 10^{-4} \mathrm m^2} = 0.21 \mathrm m \Omega $$
Now, lighting is a pretty fast pulse, so a lot of the energy will be high frequency, so our copper rod will be subject to skin effect, which will effectively increase the resistance by some factor dependent on frequency. To keep our estimate brief let's skip the full Fourier analysis on lighting and just say the resistance increases by an order of magnitude. So the resistance of a ground rod is something like $ 2 \mathrm m \Omega $.
Now according to Joule, the power in the rod is:
$$ 2\mathrm m \Omega (120 \mathrm{kA})^2 = 28800 \mathrm{kW}$$
Wow! But, lighting strikes are also really brief. A big strike transfers maybe 350 coulombs of charge. Working backwards, if the current were a constant 120kA, then to transfer that much charge would take:
$$ \frac{350 \mathrm C}{120 \mathrm{kA}} \approx 3 \mathrm {ms} $$
So all that power for that 3mS means a total energy of:
$$ 28800 \mathrm{kW} \cdot 3 \mathrm {ms} = 84 \mathrm{kJ} $$
Wolfram Alpha puts that in perspective as about the energy released by burning two grams of gasoline.
Now, you would go on to calculate the heat capacity of our grounding rod, and figure out how much hotter this energy would make the rod, and determine if that's enough to melt it. But, just through intuition and experience, I can tell you that with two grams of gasoline you can make a grounding rod pretty hot, but not melt it.
Of course, there are all kinds of effects for which we are not accounting. Lightning strikes are so fast that we must consider the inductance and thermal resistance of everything. The parts of the grounding system that carry most of the current (the surface of the rod) will get the hottest fastest. But also, the inductance of the entire grounding system helps to limit the current by spreading it over a longer time (storing the energy temporarily in the field of the inductance). And since power is proportional to the square of the current, moving the same amount of charge over a longer period of time means significantly less Joule heating in the conductors.
Also, Sears tower doesn't have just one little ground rod. It has huge steel piles that go way deeper than 8 feet and are many orders of magnitude more massive.
As far as multiple strikes accumulating to cause damage, I doubt it. Even in a big storm, the time between strikes is very long relative to the duration of the individual strikes. In this entire time, anything that got hot has time to transfer its heat to less hot things nearby. The bits in the ground have the whole Earth as a heat sink. The bits above ground are very massive.
This might seem counter-intuitive, since after all, lightning is really powerful, right? I mean, it's super bright and loud. But consider, most of the energy in the strike goes into the flash of light, and heating the air, making thunder. Though a lightning strike does indeed release a ton of energy, most of it is expended working on the air and to electromagnetic radiation. All we need to do is provide a path for the current, and it takes much less energy to move current through a metal rod than it does through air. So, the metal rod needn't absorb most of the energy.
Best Answer
A Lightning rod without earthing is as good as any other conductor laying stray in open. If its not earthed then it wont work at all. There is nothing very fancy about the whole process. You need to connect both ends of a circuit to make current flow. An improper grounding connection can cause disaster sometimes as the lightening may find another route like plumbing to reach to the ground, which will produce the same effect as that of a lightning striking any building and can result in fire. Even simple grounding is not advisable for such a system. The area at the place of grounding is treated with saline water to increase conductivity.
Many times improper grounding renders the whole system inappropriate. Even sharp curves are avoided in the conductor connecting to the ground to reduce inductance. If ever curves are made then the radius is kept large to decrease inductance.