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The period of a body under S.H.M. is represented by: T = Pa Db Sc where P is pressure, D is density and S is surfacetension, then values of a, b and c are
T =Pa Db Sc
M0 L0 T1(ML-1T-2)(ML-3)b(MT-2)c
=Ma+b+cL-a-3bT-2a-2c
Applying principle of homogeneity
a +b + c = 0; – a – 3b = 0; –2a – 2c = 1
on solving, we get a = –3/2, b = 1/2, c =1
Which physical quantities have same dimensions?
Moment of couple = force × distance[M1 L2 T-2]
work = force × distance [M1 L2 T-2].
If L denotes the inductance of an inductor through which acurrent i is flowing, the dimensions of L i2 are
Energy stored in an inductor=1⁄2 L i2=[ML2T-2]
The unit and dimensions of impedance in terms of charge Q are
Impedance=\(\frac{V}{I}=\frac{W}{QI}=\frac{ML^{2}T^{-2}}{QQT^{-1}}\)
=[ML2 T-1 Q-2]
The dimensions of Wien’s constant are
b=λmT=LK=[M0L1T0K1]
The dimensions of magnetic moment are
M= current × area = AL2 = [L2 A1]
The dimensions of universal gas constant are
\(R=\frac{PV}{\mu T}=\frac{W}{\mu T}=\frac{ML^{2}T^{-2}}{mol\, K}\)
where μ is number of mole of the gas
= [M1L2T-2K-1mol-1]
The dimensions of Rydberg’s constant are
\(From\frac{1}{\lambda }=R\left ( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right )\)
dimensions of R=1⁄L=L-1=[M0L-1T0]
The dimensions of pressure gradient are
Pressure gradient=\(\frac{Pressure\, difference}{distance}\)
[Pressure gradient] =\(\frac{ML^{-1}T^{-2}}{L}=[ML^{-2}T^{-2}]\)
What are the units of magnetic permeability?
From Biot Savart’s law
B=\(\frac{\mu _{0}}{4\pi }\frac{id\, l\, sin\, \Theta }{r^{2}}\)
\(\mu _{0}=\frac{4\pi \, Br^{2}}{id\, l\, sin\, \Theta}=\frac{Wb\, m^{-2}\, m^{2}}{Am}=Wb\, A^{-1}\, m^{-1}\)