What is meant by temperature and dimension?

Temperature defines the degree of hotness or coldness of a system. In a more precise sense, temperature designates the amount of internal energy possessed by a thermodynamic system. The internal energy of the thermodynamic system comprises molecular energy. Molecular energy is composed of molecular translational energy, vibrational energy, and rotational energy. When the sensible energy of a system increases, there is a rise in the molecular energy of the system. Hence, the consequence is a rise in the temperature of the system. The concept of temperature as a fundamental dimension is given by the zeroth law of thermodynamics, which forms the basis of temperature measurement via temperature measuring instruments like a thermometer. The unit of temperature is usually given by SI (International System of Units) base units. There are generally two SI base units of temperature- Kelvin (K) and Degree Celsius ($°C$).

In physics, dimensions are the physical quantities that can be measured. For example, in the case of temperature, the amount of molecular motion or energy is measured and expressed as the amount of heat energy possessed by the system. In science, fundamental dimensions are those dimensions that are derived from any other quantities. These are sometimes referred to as primary dimensions and form the basis of derivation of other dimensions, known as secondary dimensions.

Types of dimensions

There are two kinds of dimensions used. They are fundamental or primary dimensions and derived or secondary dimensions. The fundamental dimensions are independent quantities, whereas derived dimensions are the quantities whose base forms the fundamental quantities for their derivations.

There are seven fundamental quantities. They are- length, mass, time, temperature, luminous intensity, electric current, and amount of substance (mole). Out of which, length, mass, time, and temperature are mostly used in physical quantities in physics and engineering disciplines. Physical quantities like velocity, acceleration, work done, pressure, power, area, and density are all derived quantities. 

Definition of the dimension of a physical quantity

The dimension of a quantity can be defined as the power of the fundamental physical quantity. The dimension of the quantity is proportional to the raised powers of the fundamental physical quantity.

For example, let B be an unknown quantity. We know, there are four fundamental quantities. Hence, the general notation to denote the unknown dimension by the physical quantity is-

$B=\left[L^a M^b T^c A^d Mo{l}^e K^f C{d}^g\right]$

where $L$ denotes the dimension of length

$T$ denotes the dimension of time

$M$ denotes the dimension of mass

$A$ denotes the dimension of electric current

$Mol$ denotes the dimension of the amount of substance

$K$ denotes the dimension of temperature, and

$Cd$ denotes the dimension of luminous intensity 

$a,b,c,d,e,f,$and $g$ are the powers of the physical quantities.

Rules for writing dimensions

There are certain rules to write dimensions of a physical quantity using the fundamental dimensions. They are outlined below.

• Dimensions are to be written in enclosed brackets given by $\left[\right]$.
• When any system is independent of a fundamental quantity, its power is considered to be zero.
• While writing dimensions of a physical quantity, laws of exponents are followed.
• Plane and solid angles are dimensionless quantities. 

Dimensions of some well-known quantities

Force

The force applied to a body is the mass of the body multiplied by the acceleration, i.e. $F=m\times a$.

To obtain an expression of the dimension of force, considering the acceleration of a body, which is the change of velocity with respect to time. Mathematically, $a=\frac{∆V}{∆t}$, where $V$ denotes the velocity of the object and $t$ represents the time.

The above equation of acceleration can be written as $a=\frac{V_f-V_i}{t_f-t_i}$, where, $V_f$ and $V_i$ denote final velocity and initial velocity of a body respectively. $t_f$ and $t_i$ represent the final and initial times respectively.

Considering the body initial at rest at time $t_i=0$ and reaches a final velocity $V_f$ at time $t_f$. The above expression for acceleration then reduces to $a=\frac{V_f}{t_f} ......\left(1\right)$. 

Further, velocity is the ratio between the change of displacement with respect to time. Therefore, velocity is expressed mathematically as,

$V=\frac{∆s}{∆t}$, where $s$ represents the displacement of the body. 

The above equation can be written as $V=\frac{s_f-s_i}{t_f-t_i}$, where, $s_f$ and $s_i$ are the final and initial displacements respectively.

If a body is initially at rest and starts to move from its initial position at time $t_i=0$, and reaches a final position $s_f$ at time $t_f$, then its initial displacement can be considered zero, i.e. $s_i=0$. Hence, the expression of velocity for such a case can be written as $V=\frac{s_f}{t_f} ......\left(2\right)$.

Therefore, equation (2) can be expressed in terms of dimensional quantity as,

$V=\left[M^{0}L{T}^{-1}\right]$. Where $L$ is the dimension of length, i.e. the distance, $T$ is the dimension of time and $M$ is the dimension of mass which is zero.

Or, $V=\left[L{T}^{-1}\right] ......\left(3\right)$

Now using equation (3), the dimension of acceleration is expressed as,

$a=\left[L{T}^{-2}\right]$

Finally using the dimension of acceleration, the dimension of force, i.e. $F=m\times a$ is written as

$F=\left[ML{T}^{-2}\right]$, where $M$ denotes the dimension of mass.

Electric current

The dimension of the electric current can be written as

$I=\left[M^0{L}^0{T}^{0}A\right]$, where $I$ represents the electric current, and $A$ is the dimension of the electric current. 

Magnetic field

The magnetic field is the surrounding region of a magnet where the influence of the magnet's force field can be experienced. 

The magnetic field is expressed as

$B=Lorentz force\times {\left[charge\times velocity\right]}^{-1}$

Therefore, $B=\left[M{T}^{-2}{I}^{-1}\right]$, where $I$ represents the dimension of electric charge.

Unit system of the quantities

There are basically four systems of units to represent the physical quantities, these are outlined below.

The C.G.S. system of units

In the C.G.S. system of units, the numerical value of the fundamental quantity, length is denoted by centimeters, time is denoted by seconds, and mass by grams. 

SI based units

The word 'SI unit' means, international system of units. It is a standardized unit accepted and followed worldwide.

In the SI system of units, the fundamental quantity length is denoted by the meter ($m$), mass is denoted by the kilogram ($kg$), and time is denoted by the second ($s$). 

The unit kilogram was originally coined in 1795 to express 1 liter of water. Later in 1799, the kilogram was chosen as the standard unit of mass. 

Kilogram can be defined in terms of three fundamental constants in physics: speed of light, atomic transition frequency, and Planck's constant.

The kilogram is defined in the 18th century as the weight of solid platinum which is still used as the standardized weight.

• SI unit of volume is considered to be a cubic meter. In terms of symbols, the cubic meter is written as $m^3$.
• SI unit of area is considered to be a square meter. In terms of symbols, the square meter is written as $m^2$.  
• SI unit of temperature is considered to be Kelvin. It is written as $K$.
• SI unit of electric current is ampere. Ampere is symbolically written as $A$.
• SI unit of force is Newton. Symbolically, it is written as $N$.
• SI unit of electric field is Newtons per Coulomb. 

F. P. S. system of units

It is also sometimes referred to as a foot-pounds system of units. In this system, the length is expressed in foot $\left(f_t\right)$, mass is expressed in pounds $\left(Lb\right)$, and time is expressed in seconds $\left(s\right)$. 

Dimensionless quantities

Dimensionless quantities are those quantities that have no dimensions. They are also known as bare, pure, and scalar quantities. Dimensionless quantities have a dimension of unity, or 1.

Angle is one such example of dimensionless quantity.

The unit of plane angle is Radian and the unit of solid angle is Steradian.

Angles can also be expressed in degrees, but for it, proper unit conversion needs to be followed using conversion factors.

Context and Applications

This topic is primarily taught in many undergraduate and postgraduate degree courses of:

• Bachelors in Science in Physics
• Bachelors in Technology in Mechanical Engineering
• Bachelors in Technology in Civil Engineering
• Masters in Science in Physics

Practice Problems

1. Which of the following laws of thermodynamics denotes temperature measurement?

1. Zeroth law
2. First law
3. Second law
4. Fourth law

Answer: Option a

Explanation: The zeroth law of thermodynamics denotes temperature measurement.

2. What is the SI unit of force?

1. Newton
2. Pound-force
3. Foot-pounds
4. None of these

Answer: Option a

Explanation: The SI unit of force is Newton.

3. Which of the following is a dimensionless quantity?

1. Velocity
2. Acceleration
3. Angle
4. Force

Answer: Option c

Explanation: Angle is a dimensionless quantity.

4. Which of the following is responsible for temperature rise in a system?

1. Potential energy
2. Latent energy
3. Internal energy
4. Kinetic energy

Answer: Option c

Explanation: Internal energy is responsible for the rise in temperature of a system.

5. What is the SI unit of the electric field?

1. Coulomb
2. Candela
3. Newtons per Coulomb
4. None of these

Answer: Option c

Explanation: The SI unit of the electric field is Newtons per Coulomb.