In the eqn.\(\left ( p+\frac{a}{V^{2}} \right )\)(V-b)=constant,the unit of a is
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Solution
As \(\frac{a}{V^{2}}=P\)
∴ a=\(PV^{2}=\frac{dyne}{cm^{2}}(cm^{3})^{2}\)=dyne × cm4
The time dependence of a physical quantity p is given by p = p0 exp.(– α t2), where is a constant and t is the time.The constant α
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Solution
In ρ = p0 exp. (– α t2) is dimensionless
∴ α=\(\frac{1}{t^{2}}=\frac{1}{T^{2}}=[T^{-2}]\)
The velocity of water waves (v) may depend on their wavelength λ, the density of water ρ and the acceleration due to gravity, g. The method of dimensions gives there lation between these quantities is
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Solution
v=k λa ρb gc
[M0LT-1]=La (ML-3)b(L T-2)c
=Mb La-3b+c T-2c
∴ b = 0; a - 3b + c = 1
-2c=-1⇒ c=1/2 ∴a=1⁄2
v α λ1/2 ρ0g1/2 or v2 α λ g
If e is the charge, V the potential difference, T the temperature,then the units of \(\frac{eV}{T}\) are the same as that of
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Solution
\(\frac{eV}{T}=\frac{W}{T}=\frac{PV}{T}=R\)
and R⁄N= Boltzmann constant
The physical quantity which has the dimensional formula[M1T-3] is
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Solution
Solar constant = energy/area/time
\(\frac{ML^{2}T^{-2}}{L^{2}T}=[M^{1}T^{-3}]\)
L, C, Rrepresent physical quantities inductance, capacitance and resistance respectively. The combinations which have the dimensions off requency are
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Solution
\(\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{(ML^{2}T^{-2}A^{-2})\times (M^{-1}L^{-2}T^{4}A^{2})}}\)
\(\frac{1}{T^{2}}=T^{-1}\)
Dimensions of specific heat are
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Solution
s=\(\frac{Q}{m\theta }=\frac{ML^{2}T^{-2}}{MK}\)=[L2T2K-1]
The dimensional formula for entropy is
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Solution
Entropy=\(\frac{Q}{T}=\frac{ML^{2}T^{-2}}{K}\)=[ML2T-2K-1]
The dimensions of \(\frac{1}{\epsilon _{0}}\frac{e^{2}}{hc}\) are
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Solution
From F=\(\frac{1}{4\pi \varepsilon _{0}}\frac{e^{2}}{r^{2}}\)
∴\(\frac{e^{2}}{\varepsilon _{0}}=4\pi Fr^{2}\)(dimensionally)
\(\frac{e^{2}}{\varepsilon _{0}hc}=\frac{4\pi Fr^{2}}{hc}=\frac{(MLT^{-2})L^{2}}{ML^{2}T^{-1}[LT^{-1}]}\)=[M0L0T0A0]
\(\frac{e^{2}}{\varepsilon _{0}hc}\) is called fine structure constant & has value\(\frac{1}{137}\)
The dimensions of solar constant is
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Solution
Solar constant = energy/sec/area
\(\frac{mL^{2}T^{-2}}{TL^{2}}\)=MT-3