Question: Which of the following physical quantities has neither dimensions nor unit? A. angle B. luminous...

Question

Which of the following physical quantities has neither dimensions nor unit?
A. angle
B. luminous intensity
C. coefficient of friction
D. current

Answer 1

Solution

Some of the options like angle is having a unit even though it is dimensionless. If not a base quantity, use the formula that describes the quantity. Find out the dimensions of each quantity then determine whether it is dimensionless or not.

Complete step by step answer:
First of all let us check the dimensions of each of the quantities given in the question.
Angle is described as the ratio of the arc length to the radius. This can be written in the dimensional formula as,
$\theta =\dfrac{\left[ L \right]}{\left[ L \right]}=\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}} \right]$
From this we can see that the angle is dimensionless. However it has units as radians.

Next one is luminous intensity which is given as the amount of the power radiated by a light in a specific direction divided by unit solid angle.
The dimensional formula will be,
Light energy emitted per second = $\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{T}$
Solid angle is dimensionless quantity.
Therefore, luminous intensity will be,
$J=\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{T}=\left[ M{{L}^{2}}{{T}^{-3}} \right]$
Its unit is given as candela.

Current is the flow of charge divided by time.
This can be written as the equation,
$I=\dfrac{q}{t}$
This is a fundamental quantity with a dimension of $\left[ I \right]=\left[ A \right]$
The unit of current is Ampere.

Lastly, coefficient of friction. This is a constant in the frictional force relationship.
$\mu =\dfrac{F}{N}$
It is the ratio of two forces, therefore it does not have a dimension as well as a unit.
$\left[ \mu \right]=\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}} \right]$

So, the correct answer is “Option C”.

Note: A coefficient of friction is the measure of the strength of frictional force. If it is higher, then the frictional force also will be higher than the normal force. Silicone rubber is an example for a higher coefficient of friction which is greater than one.