Unit and Dimensions: Problems

1. In SI system, the unit of Temperature is

    (a) Degree Celcius

    (b) Degree Centigrade

    (c) Degree Kelvin

    (d) Degree Fahrenheit 

[d]

2. Which of the following is a unit of distance?

     (a) Metre

     (b) Astronomical unit

     (c) Light year

     (d)  All of the above

[d]

3. What is the dimension of surface tension:

     (a) $[M^2L^2T^{-2}$

     (b) $[M^2LT^{-2}]$

     (c) $[ML^0T^{-2}]$

     (d) None of these

[c]

4. Which of the following have the same dimensions?

     (a) Stress

     (b) Bulk modulus

     (c) Thrust

     (d) Energy Density

[a and b]

5. If C and R denote the capacitance and resistance, then the dimension of RC is:

     (a) $[M^0L^0T ]$

     (b) $[ML^0T]$

     (c)  $[MLT]$

     (d)  $[M^0L^0T^2]$

[a]

6. Write any two physical quantities which have the dimension of Energy? [Torque, Work]

7. Is it possible to add any two physical quantities? [No]

8. Force on a sphere of radius 'a' moving in the medium with velocity 'v' is given by $F = 6\pi \eta av$. Find the dimension of $\eta$. [$ML^{-1}T^{-1}$]

9. What is the dimensional formula for Planck's constant? [$ML^2T^{-1}$]

10. In the formula, $X = 3YZ^2$, X and Z have the dimensions of capacitance and magnetic induction respectively. Find the dimension of Y in MKSQ system. [$M^{-3}L^{-2}T^{4}Q^{4}$]

11. The equation of state for a real gas is given by $(p+\frac{a}{V^2})(v-b) = RT$. Find the dimensions of 'a' and 'b'. [$[ML^5T^{-2}]$, $[L^3]$]

12. Find the dimension of $\frac{1}{2}\epsilon_0 E^2$. [$ML^{-1}T^{-2}$]

13. A quantity X is given by $\epsilon_0 L\frac{\Delta V}{\Delta t}$ where $\epsilon_0$ is the permittivity of the free space, L is the length,  $\Delta$V is the potential difference and ${\Delta t}$ is a time interval. The dimensional formula for X is the same as that of  (a) Resistance (b) Charge (c) Voltage (d) Current. [d]

14. Pressure depends on distance as $p = \frac{\alpha}{\beta}e^{-\frac{\alpha z}{k \theta}}$, where $\alpha$, $\beta$ are constant, z is the distance, k is Boltzman's constant and $\theta$ is temperature. Find the dimension of $\beta$. [$L^{2}$]

15. Which of the following pair (s) has the same dimension?

     (a) Torque and work

     (b) Angular momentum and Work

     (c) Energy and Young's modulus

     (d) Light-year and Wavelength 

     (e) Reynold number and co-efficient of friction

     (f) Curie and Frequency of the light wave

     (g) Latent heat and gravitational potential

     (h) Planck's constant and torque

[a,d,e,f,g]

16. Let $[\epsilon_0]$ denote the dimensional formula of permittivity of the vacuum, and $[\mu_0]$ that of permeability of the vacuum. Find their dimensional formula in term of mass M, length L, time T, and electric current I. [$\epsilon_0 = [M^{-1}L^{-3}T^4I^2]$, $\mu_0= [MLT^{-2}I^{-2}]$]

17. Planck's constant h, speed of light c, and gravitational constant G are used to form a unit of length L and a unit of mass M. Then find the correct option (s) is (are): 

   (a) $M \propto \sqrt{c}$ 

   (b) $M \propto \sqrt{G}$ 

   (c) $L \propto \sqrt{h}$ 

   (d) $L \propto \sqrt{G}$

[a,c,d]

18. In term of potential difference V, electric current I, permittivity $\epsilon_0$, permeability $\mu_0$ and speed of light c, the dimensionally correct equation(s) is(are):

    (a) $\mu_0 I^2 = \epsilon_0V^2$ 

    (b) $\mu_0 I = \epsilon_0V$ 

    (c) $I = \epsilon_0 cV$ 

    (d) $\mu_0 c I = \epsilon_0 V$ 

[a,c]

19. A length-scale (l) depends on the permittivity ($\epsilon$) of a dielectric material. Boltzmann constant ($k_B$), the absolute temperature (T), the number per unit volume (n) of certain charged particles, and charge (q) carried by each of the particles. Which of the following expression(s) for l is/are dimensionally correct? 

    (a) $l = \sqrt{\frac{nq^2}{\epsilon k_B T}}$ 

    (b) $l = \sqrt{\frac{\epsilon k_B T}{nq^2}}$ 

    (c) $l = \sqrt{\frac{q^2}{\epsilon n^{2/3}k_B T}}$ 

    (d) $l = \sqrt{\frac{q^2}{\epsilon n^{1/3}k_B T}}$

[b,d]

20. Give the MKS unit of each of the following: 

    (a) Young's Modulus

    (b) Magnetic Induction

    (c) Power of lens

[$N/m^2$, Tesla, Dioptre]

21. A gas bubble, from an explosion underwater, oscillate with a period T proportional to $p^ad^bE^c$, where  'P' is the static pressure, 'd' is the density of the water, and 'E' is the total energy of the explosion. Find the values of a, b, and c. [$a = -\frac{5}{6},b=\frac{1}{2},c=\frac{1}{3}$]

22. Write the dimensions of the following in terms of mass, time, length, and charge

    (a) Magnetic flux 

    (b) Rigidity modulus

[$[ML^2T^{-1}Q^{-1}], [ML^{-1}T^{-2}]$]

23. Match the following with their dimensions where Q is for charge:

Column I	Column II
(A) Angular momentum	(a) $[ML^2T^{-2}]$
(B) Latent Heat	(b) $[ML^2Q^{-2}]$
(C) Torque	(c) $[ML^2T^{-1}]$
(D) Capacitance	(d) $[ML^3T^{-1}Q^{-2}]$
(E) Inductance	(e) $[M^{-1}L^{-2}T^{2}Q^2]$
(F) Resistivity	(f) $[L^2T^{-2}]$

[$(A) \to (c),(B) \to (f) ,(C)\to (a), (D)\to (e), (E)\to (b),(F)\to (d)$]

24. Match column I with column II:

Column I	Column II
(A) Capacitance	(i)                  Ohm-second
(B) Inductance	(ii)                Coulomb2-joule-1
(C) Magnetic Induction	(iii)               Coulomb (volt)-1
	(iv)               Newton (amp-metre)-1
	(v)                Volt-second (ampere)-1

[$(A) \to (ii),(iii), (B)\to (i),(v), (C)\to (iv)$]