If e is the electronic charge, c is the speed of light in free space and h is Planck's constant, the quantity {1 \over {4\pi {\varepsilon _0}}}{{|e{|^2}} \over {hc}} has dimensions of :

In a typical combustion engine the workdone by a gas molecule is given by W = {\alpha ^2}\beta {e^{{{ - \beta {x^2}} \over {kT}}}}, where x is the displacement, k is the Boltzmann constant and T is the temperature. If $$\alpha$$ and $$\beta$$ are constants, dimensions of $$\alpha$$ will be :

If 'C' and 'V' represent capacity and voltage respectively then what are the dimensions of $$\lambda$$ where C/V = $$\lambda$$ ?

One main scale division of a vernier callipers is 'a' cm and nth division of the vernier scale coincide with (n $$-$$ 1)th division of the main scale. The least count of the callipers in mm is :

In order to determine the Young's Modulus of a wire of radius 0.2 cm (measured using a scale of least count = 0.001 cm) and length 1m (measured using a scale of least count = 1 mm), a weight of mass 1 kg (measured using a scale of least count = 1 g) was hanged to get the elongation of 0.5 cm (measured using a scale of least count 0.001 cm). What will be the fractional error in the value of Young's Modulus determined by this experiment?

The vernier scale used for measurement has a positive zero error of 0.2 mm. If while taking a measurement it was noted that '0' on the vernier scale lies between 8.5 cm and 8.6 cm, vernier coincidence is 6, then the correct value of measurement is ___________ cm. (least count = 0.01 cm)

If time (t), velocity (v), and angular momentum (l) are taken as the fundamental units. Then the dimension of mass (m) in terms of t, v and l is :

The force is given in terms of time t and displacement x by the equation F = A cos Bx + C sin Dt The dimensional formula of {{AD} \over B} is :

Assertion A : If in five complete rotations of the circular scale, the distance travelled on main scale of the screw gauge is 5 mm and there are 50 total divisions on circular scale, then least count is 0.001 cm. Reason R : Least Count = {{Pitch} \over {Total\,divisions\,on\,circular\,scale}} In the light of the above statements, choose the most appropriate answer from the options given below :

A physical quantity 'y' is represented by the formula y = {m^2}{r^{ - 4}}{g^x}{l^{ - {3 \over 2}}} If the percentage errors found in y, m, r, l and g are 18, 1, 0.5, 4 and p respectively, then find the value of x and p.

In a Screw Gauge, fifth division of the circular scale coincides with the reference line when the ratchet is closed. There are 50 divisions on the circular scale, and the main scale moves by 0.5 mm on a complete rotation. For a particular observation the reading on the main scale is 5 mm and the 20th division of the circular scale coincides with reference line. Calculate the true reading.

If E, L, M and G denote the quantities as energy, angular momentum, mass and constant of gravitation respectively, then the dimensions of P in the formula P = EL2M$$-$$5G$$-$$2 are :

Match List - I with List - II Choose the most appropriate answer from the options given below :

If the length of the pendulum in pendulum clock increases by 0.1%, then the error in time per day is :

Which of the following is not a dimensionless quantity?

If E and H represents the intensity of electric field and magnetising field respectively, then the unit of E/H will be :

Match List - I with List - II. List - I List - II (a) $$R_H$$ (Rydberg constant) (i) $$kg{m}^{-1}{s}^{-1}$$ (b) h (Planck's constant) (ii) $$kg{m}^2{s}^{-1}$$ (c) $${\mu }_B$$ (Magnetic field energy density) (iii) $$m^{-1}$$ (d) $$\eta$$ (coefficient of viscocity) (iv) $$kg{m}^{-1}{s}^{-2}$$ Choose the most appropriate answer from the options given below :

If force (F), length (L) and time (T) are taken as the fundamental quantities. Then what will be the dimension of density :

Match List - I with List - II. List - I List - II (a) Torque (i) MLT$${}^{-1}$$ (b) Impulse (ii) MT$${}^{-2}$$ (c) Tension (iii) ML$${}^2$$T$${}^{-2}$$ (d) Surface Tension (iv) MLT$${}^{-2}$$ Choose the most appropriate answer from the option given below :

Which of the following equations is dimensionally incorrect? Where t = time, h = height, s = surface tension, $$\theta$$ = angle, $$\rho$$ = density, a, r = radius, g = acceleration due to gravity, v = volume, p = pressure, W = work done, T = torque, $$\in$$ = permittivity, E = electric field, J = current density, L = length.