PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions

Find the dimensions of magnetic permeability \[\mu_0\] 
The relevant equation are \[F = qE, F = qvB, \text{ and }B = \frac{\mu_0 I}{2 \pi a};\]

Find the dimensions of Planck's constant h from the equation E = hv where E is the energy and v is the frequency.

Find the dimensions of the specific heat capacity c.
(a) the specific heat capacity c,
(b) the coefficient of linear expansion α and
(c) the gas constant R.
Some of the equations involving these quantities are \[Q = mc\left( T_2 - T_1 \right), l_t = l_0 \left[ 1 + \alpha\left( T_2 - T_1 \right) \right]\] and PV = nRT.

Find the dimensions of the coefficient of linear expansion α and

The height of mercury column in a barometer in a Calcutta laboratory was recorded to be 75 cm. Calculate this pressure in SI and CGS units using the following data : Specific gravity of mercury = \[13 \cdot 6\] , Density of \[\text{ water} = {10}^3 kg/ m^3 , g = 9 \cdot 8 m/ s^2\] at Calcutta. Pressure
= hpg in usual symbols.

Theory of relativity reveals that mass can be converted into energy. The energy E so obtained is proportional to certain powers of mass m and the speed c of light. Guess a relation among the quantities using the method of dimensions.

Let I = current through a conductor, R = its resistance and V = potential difference across its ends. According to Ohm's law, product of two of these quantities equals the third. Obtain Ohm's law from dimensional analysis. Dimensional formulae for R and V are \[{\text{ML}}^2 \text{I}^{- 2} \text{T}^{- 3}\] and \[{\text{ML}}^2 \text{T}^{- 3} \text{I}^{- 1}\] respectively.

Test if the following equation is dimensionally correct:
\[h = \frac{2S cos\theta}{\text{ prg }},\]
where h = height, S = surface tension, ρ = density, I = moment of interia.

Test if the following equation is dimensionally correct:
\[v = \sqrt{\frac{P}{\rho}},\]

where v = velocity, ρ = density, P = pressure

Test if the following equation is dimensionally correct:
\[V = \frac{\pi P r^4 t}{8 \eta l}\]

where v = frequency, P = pressure, η = coefficient of viscosity.

Test if the following equation is dimensionally correct:
\[v = \frac{1}{2 \pi}\sqrt{\frac{mgl}{I}};\] 
where h = height, S = surface tension, \[\rho\] = density, P = pressure, V = volume, \[\eta =\] coefficient of viscosity, v = frequency and I = moment of interia.

Let x and a stand for distance. Is
\[\int\frac{dx}{\sqrt{a^2 - x^2}} = \frac{1}{a} \sin^{- 1} \frac{a}{x}\] dimensionally correct?

Can you add three unit vectors to get a unit vector? Does your answer change if two unit vectors are along the coordinate axes?

Can you add two vectors representing physical quantities having different dimensions? Can you multiply two vectors representing physical quantities having different dimensions?

Let ε₁ and ε₂ be the angles made by  \[\vec{A}\] and -\[\vec{A}\] with the positive X-axis. Show that tan ε₁ = tan ε_2. Thus, giving tan ε does not uniquely determine the direction of \[\vec{A}\].

Is the vector sum of the unit vectors  \[\vec{i}\] and \[\vec{i}\] a unit vector? If no, can you multiply this sum by a scalar number to get a unit vector?

Let \[\vec{A} = 3 \vec{i} + 4 \vec{j}\]. Write a vector \[\vec{B}\] such that \[\vec{A} \neq \vec{B}\], but A = B.