A force $F$ is given by $F = at + b{t^2}$, where $t$ is time. What are the dimensions of $a$ and $b$
A force $F$ is given by $F = at + b{t^2}$, where $t$ is time. What are the dimensions of $a$ and $b$
- A
$ML{T^{ - 3}}$ and $M{L^2}{T^{ - 4}}$
- B
$ML{T^{ - 3}}$ and $ML{T^{ - 4}}$
- C
$ML{T^{ - 1}}$ and $ML{T^0}$
- D
$ML{T^{ - 4}}$ and $ML{T^1}$
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A length-scale $(l)$ depends on the permittivity $(\varepsilon)$ of a dielectric material. Boltzmann constant $\left(k_B\right)$, the absolute temperature $(T)$, the number per unit volune $(n)$ of certain charged particles, and the charge $(q)$ carried by each of the particless. Which of the following expression($s$) for $l$ is(are) dimensionally correct?
($A$) $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$
($B$) $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$
($C$)$l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$
($D$) $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$
A length-scale $(l)$ depends on the permittivity $(\varepsilon)$ of a dielectric material. Boltzmann constant $\left(k_B\right)$, the absolute temperature $(T)$, the number per unit volune $(n)$ of certain charged particles, and the charge $(q)$ carried by each of the particless. Which of the following expression($s$) for $l$ is(are) dimensionally correct?
($A$) $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$
($B$) $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$
($C$)$l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$
($D$) $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$
- [IIT 2016]
The velocity of water waves $v$ may depend upon their wavelength $\lambda $, the density of water $\rho $ and the acceleration due to gravity $g$. The method of dimensions gives the relation between these quantities as
The velocity of water waves $v$ may depend upon their wavelength $\lambda $, the density of water $\rho $ and the acceleration due to gravity $g$. The method of dimensions gives the relation between these quantities as