Write the dimensions of $$a \times b$$ in the relation $$E = \frac{{b - {x^2}}}{{at}},$$ Where $$E$$ is the energy, $$x$$ is the displacement and $$t$$ is time
A.
$$M{L^2}T$$
B.
$${M^{ - 1}}{L^2}{T^1}$$
C.
$$M{L^2}{T^{ - 2}}$$
D.
$$ML{T^{ - 2}}$$
Answer :
$${M^{ - 1}}{L^2}{T^1}$$
Solution :
Here, $$b$$ and $${x^2} = {L^2}$$ have same dimensions
Also, $$a = \frac{{{x^2}}}{{E \times t}} = \frac{{{L^2}}}{{\left( {M{L^2}{T^{ - 2}}} \right)T}} = {M^{ - 1}}{T^1}$$
$$a \times b = \left[ {{M^{ - 1}}{L^2}{T^1}} \right]$$
Releted MCQ Question on Basic Physics >> Unit and Measurement
Releted Question 1
The dimension of $$\left( {\frac{1}{2}} \right){\varepsilon _0}{E^2}$$ ($${\varepsilon _0}$$ : permittivity of free space, $$E$$ electric field)
A quantity $$X$$ is given by $${\varepsilon _0}L\frac{{\Delta V}}{{\Delta t}}$$ where $${ \in _0}$$ is the permittivity of the free space, $$L$$ is a length, $$\Delta V$$ is a potential difference and $$\Delta t$$ is a time interval. The dimensional formula for $$X$$ is the same as that of-
Pressure depends on distance as, $$P = \frac{\alpha }{\beta }exp\left( { - \frac{{\alpha z}}{{k\theta }}} \right),$$ where $$\alpha ,$$ $$\beta $$ are constants, $$z$$ is distance, $$k$$ is Boltzman’s constant and $$\theta $$ is temperature. The dimension of $$\beta $$ are-