A metallic block has no potential difference applied across it, then the mean velocity of free electrons at absolute temperature T is
A. Proportional to T
B. Proportional to \[\sqrt T \]
C. Zero
D. Finite but independent of T

Hint: The concept used in this problem is the kinetic theory of gases and the Maxwell-Boltzmann distribution. We will use the Maxwell-Boltzmann distribution to find the mean velocity of free electrons in a metallic block at a given temperature.

Complete step by step solution:
We know that in a metallic block, the conduction of electricity is mainly due to the movement of free electrons. According to the kinetic theory of gases, the velocity distribution of free electrons in a metallic block follows the Maxwell-Boltzmann distribution. The Maxwell-Boltzmann distribution states that the probability of finding an electron with a velocity v is given by
$P(v) = \left(\dfrac{m}{2\pi kT}\right)^{3/2} v^2 e^{-\dfrac{mv^2}{2kT}}$

Now, using this distribution, we can find the mean velocity of free electrons as,
$\bar{v} = \sqrt{\dfrac{8kT}{\pi m}}$
Hence, we can see that the mean velocity of free electrons is directly proportional to $\sqrt T$.

Therefore, the correct option is B.

Note: The Maxwell-Boltzmann distribution describes the velocity distribution of particles in a gas. It is a statistical law that describes the probability of finding a particle with a certain velocity at a given temperature. A Point should be noted that, When all parameters are held equal, the greater the potential difference in the system, the greater the velocity of electrons flowing in a given material will be. The scalar quantity, in the situation of irrotational flow, whose negative gradient matches the velocity is the velocity potential.