Question

The dimensional formula of the Hall coefficient is:
A) ${M^0}{L^0}{A^{ - 2}}{T^{ - 1}}$
B) ${M^0}{L^3}AJ$
C) ${M^0}{L^0}AT$
D) ${M^0}{L^3}{A^{ - 1}}{T^{ - 1}}$

Answer 1

The Hall coefficient is $r = \dfrac{1}{{nq}}$ [Where $n$ is the number density of charge and $q$ is the value of charge]. Again, $n = \dfrac{N}{V}$. Volume is length, multiplied with breadth, multiplied with height and charge is the product of electric current and time.

Complete step by step solution:
In electrostatics, the Hall coefficient is the ratio of the induced electric field to the product of the applied magnetic field, and the current density. The value of this Hall coefficient depends on the type, number, and properties of the charge carriers, which constitute the current and hence, the Hall coefficient defines the characteristic of the material from which the conductor is made.
Let, the Hall coefficient is $r$. So, $r = \dfrac{1}{{nq}}$ [Where, $n$ is the number density of charge and $q$ is the value of charge] Now, the number density of charge is the amount of electric charge per unit length. Surface area, or volume. Let, $N$ is the number of charges and $V$ is the volume. So, $n = \dfrac{N}{V}$ So, $r = \dfrac{1}{{\dfrac{{Nq}}{V}}} = \dfrac{V}{{Nq}}$ Now, $N$ has no unit. Volume is length, multiplied with breadth, multiplied with height, i.e., $V = length \times breadth \times height$. Breadth and height are quantities equivalent to length. Each has a dimension $L$. So, the dimension of the volume is $L \times L \times L = {L^3}$ The charge is the product of electric current and time. So, the dimension of charge is $AT$. So, the dimension of the Hall coefficient is $\dfrac{{{L^3}}}{{AT}} = {L^3}{A^{ - 1}}{T^{ - 1}}$ As mass $M$ has no effect on it, it is considered as ${M^0}$, which means $1$. So, the dimension of the Hall coefficient is ${M^0}{L^3}{A^{ - 1}}{T^{ - 1}}$.

Note: The dimension of a physical quantity is its relationship with the seven quantities, each of which has been assigned, by convention, a base unit. These seven quantities are 1) length $\left( L \right)$, 2) mass $\left( M \right)$, 3) time $\left( T \right)$, 4) electric current $\left( A \right)$, 5) thermodynamic temperature $\left( \theta \right)$, 6) amount of substance $\left( n \right)$, and 7) luminous intensity $\left( {{I_v}} \right)$.