Question

Which of the following is a unit less and dimensionless quantity?
A. Angle
B. solid angel
C. mechanical equivalent of heat
D. refractive index

Answer 1

Hint: We know from dimensional analysis, a physical quantity which has no dimension is called a dimensionless quantity. It is also known as a scalar quantity. This is often seen when taking the ratio between two identical dimensions or similar dimensions.

Complete step by step answer:
In physics however, there are many dimensionless quantities. These are often used as proportionality constants. For example a include;
1. The fine structure is a constant which gives the magnitude of the interactions between the electrons.
2. The Planck’s constant, which gives the ratio of the mass between the given particle and the elementary particles.
To begin with let us consider the definition of the following quantities.
a. Angle is a geometrical figure which is formed when two lines or rays meet. It has units radians or degrees. Clearly A has units and is not the answer.
b. Solid angle is the surface area swept by a circle of radius r, and it is the ratio between the surface area and the radius of the sphere. It has units similar to that of the angle, i.e. radians or degree. Clearly B has units and is not the answer.
c. Mechanical equivalent of heat is the energy of the system is conserved, and then the work done on the system is converted to heat energy. Clearly it is from heat and has units joules or calories. Clearly C has units and is not the answer.

d. Refractive index is the property of a material to bend light when passing through the medium. It is taken as the ratio of the speed of light in vacuum to the speed of the light in the given medium. Since it is the ratio of speed, it has no dimension or units. Then clearly D is the answer.

So, the correct answer is “Option D”.

Note: Dimensionless quantities are mostly obtained when the ratio of the dimensions of two quantities cancels each other or the quantities are themselves dimensionless. Thus the units also cancel each other. Hence a dimensionless quantity also has no units.