The fundamental constants are given to be permittivity constant $\left(\epsilon_0\right)$, Planck's ...

To determine which quantity constructed from the given fundamental constants has the dimensions of an electric charge, let's start by identifying the dimensions of each constant and then evaluate each expression.We have the following fundamental constants:Permittivity constant $\epsilon_0$:Derived from Coulomb's law: $F = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2}$.Dimensions: $[\epsilon_0] = [M^{-1} L^{-3} T^4 I^2]$.Planck's constant $h$:Derived from the energy formula: $E = h\nu$.Dimensions: $[h] = [M L^2 T^{-1}]$.Velocity of light $c$:As $c$ represents a speed, the dimensions are: $[c] = [L T^{-1}]$.Newton's constant $G$:Derived from the gravitational force formula: $F = G \frac{m_1 m_2}{r^2}$.Dimensions: $[G] = [M^{-1} L^3 T^{-2}]$.Boltzmann's constant $k_B$:Derived from the relation $E = k_B T$.Dimensions: $[k_B] = [M L^2 T^{-2} \Theta^{-1}]$.Now, let's check each option to see if it results in the dimensions of electric charge, which is $[I T]$:Option A:$\sqrt{\frac{h G}{\epsilon_0^3}}$$\sqrt{\frac{\left[L^5 T^{-3}\right]}{\left[M^{-3} L^{-9} T^{12} I^6\right]}} = [M^{3/2} L^7 T^{-15/2} I^{-3}]$.Option B:$\sqrt{h c}$$[\sqrt{h c}] = \sqrt{[M L^2 T^{-1}] [L T^{-1}]} = \sqrt{[M L^3 T^{-2}]} = [M^{1/2} L^{3/2} T^{-1}]$.Option C:$k_B \sqrt{h c}$$[k_B \sqrt{h c}] = [M L^2 T^{-2} \Theta^{-1}] [M^{1/2} L^{3/2} T^{-1}] = [M^{3/2} L^{7/2} T^{-3} \Theta^{-1}]$.Option D:$\sqrt{\epsilon_0 h c}$$[\sqrt{\epsilon_0 h c}] = \sqrt{[M^{-1} L^{-3} T^4 I^2] [M L^2 T^{-1}] [L T^{-1}]} = \sqrt{[T^2 I^2]} = [I T]$.Upon evaluating each option, onlyOption D: $\sqrt{\epsilon_0 h c}$has the dimensions of an electric charge $[I T]$.

Two fixed point particles, each of charge $+q$, are separated by a distance $2 L$. Another point cha...

To determine the expression for $\alpha$ in SI units, use the given relation:$ T = 2 \pi^{3/2} \alpha \frac{\sqrt{m}}{q} $From this, the dimensional formula for $\alpha$ is:$ [\alpha] = \frac{[T][q]}{[\sqrt{m}]} $Substituting the dimensional formulas, we have:$ [\alpha] = \frac{[T][I \cdot T]}{\left[M^{1/2}\right]} = \left[M^{-1/2} L^0 T^2 I^1\right] $Next, consider the dimensions of the permittivity of free space $\epsilon_0$, which can be derived from Coulomb's law:$ F = \frac{1}{4 \pi \epsilon_0} \frac{q^2}{r^2} $The dimensional formula for $\epsilon_0$ is:$ [\epsilon_0] = \left[M^{-1} L^{-3} T^4 I^2\right] $Taking the square root of $\epsilon_0$, we get:$ [\epsilon_0]^{1/2} = \left[M^{-1/2} L^{-3/2} T^2 I^1\right] $By comparing the dimensions of $\epsilon_0^{1/2}$ with $\alpha$, it is apparent that $\epsilon_0^{1/2}$ matches the dimensions of $\alpha$ except for a factor of $L^{3/2}$. Thus, the expression for $\alpha$ in terms of $\epsilon_0$ and $L$ is:$ \alpha = \epsilon_0^{1/2} L^{3/2} $

A quantity has been measured to have a value of 0.00230 in some units. How many significant figures ...

The measured value $0.00230$ contains three significant figures:The zeros before the digit 2 are not significant as they only help to place the decimal point.The digits 2 and 3 are significant.The trailing zero after the digit 3 is also significant because it is located after the decimal point and follows a non-zero digit.

Units And Measurement And Dimensions - IAT-IISER Physics PYQs

To determine which quantity constructed from the given fundamental constants has the dimensions of an electric charge, let's start by identifying the dimensions of each constant and then evaluate each expression.We have the following fundamental constants:Permittivity constant

%5Cepsilon_0

:Derived from Coulomb's law:

F%20%3D%20%5Cfrac%7B1%7D%7B4%20%5Cpi%20%5Cepsilon_0%7D%20%5Cfrac%7Bq_1%20q_2%7D%7Br%5E2%7D

.Dimensions:

%5B%5Cepsilon_0%5D%20%3D%20%5BM%5E%7B-1%7D%20L%5E%7B-3%7D%20T%5E4%20I%5E2%5D

.Planck's constant

h

:Derived from the energy formula:

E%20%3D%20h%5Cnu

.Dimensions:

%5Bh%5D%20%3D%20%5BM%20L%5E2%20T%5E%7B-1%7D%5D

.Velocity of light

c

:As

c

represents a speed, the dimensions are:

%5Bc%5D%20%3D%20%5BL%20T%5E%7B-1%7D%5D

.Newton's constant

G

:Derived from the gravitational force formula:

F%20%3D%20G%20%5Cfrac%7Bm_1%20m_2%7D%7Br%5E2%7D

.Dimensions:

%5BG%5D%20%3D%20%5BM%5E%7B-1%7D%20L%5E3%20T%5E%7B-2%7D%5D

.Boltzmann's constant

k_B

:Derived from the relation

E%20%3D%20k_B%20T

.Dimensions:

%5Bk_B%5D%20%3D%20%5BM%20L%5E2%20T%5E%7B-2%7D%20%5CTheta%5E%7B-1%7D%5D

.Now, let's check each option to see if it results in the dimensions of electric charge, which is

%5BI%20T%5D

:Option A:

%5Csqrt%7B%5Cfrac%7Bh%20G%7D%7B%5Cepsilon_0%5E3%7D%7D

%5Csqrt%7B%5Cfrac%7B%5Cleft%5BL%5E5%20T%5E%7B-3%7D%5Cright%5D%7D%7B%5Cleft%5BM%5E%7B-3%7D%20L%5E%7B-9%7D%20T%5E%7B12%7D%20I%5E6%5Cright%5D%7D%7D%20%3D%20%5BM%5E%7B3%2F2%7D%20L%5E7%20T%5E%7B-15%2F2%7D%20I%5E%7B-3%7D%5D

.Option B:

%5Csqrt%7Bh%20c%7D

%5B%5Csqrt%7Bh%20c%7D%5D%20%3D%20%5Csqrt%7B%5BM%20L%5E2%20T%5E%7B-1%7D%5D%20%5BL%20T%5E%7B-1%7D%5D%7D%20%3D%20%5Csqrt%7B%5BM%20L%5E3%20T%5E%7B-2%7D%5D%7D%20%3D%20%5BM%5E%7B1%2F2%7D%20L%5E%7B3%2F2%7D%20T%5E%7B-1%7D%5D

.Option C:

k_B%20%5Csqrt%7Bh%20c%7D

%5Bk_B%20%5Csqrt%7Bh%20c%7D%5D%20%3D%20%5BM%20L%5E2%20T%5E%7B-2%7D%20%5CTheta%5E%7B-1%7D%5D%20%5BM%5E%7B1%2F2%7D%20L%5E%7B3%2F2%7D%20T%5E%7B-1%7D%5D%20%3D%20%5BM%5E%7B3%2F2%7D%20L%5E%7B7%2F2%7D%20T%5E%7B-3%7D%20%5CTheta%5E%7B-1%7D%5D

.Option D:

%5Csqrt%7B%5Cepsilon_0%20h%20c%7D

%5B%5Csqrt%7B%5Cepsilon_0%20h%20c%7D%5D%20%3D%20%5Csqrt%7B%5BM%5E%7B-1%7D%20L%5E%7B-3%7D%20T%5E4%20I%5E2%5D%20%5BM%20L%5E2%20T%5E%7B-1%7D%5D%20%5BL%20T%5E%7B-1%7D%5D%7D%20%3D%20%5Csqrt%7B%5BT%5E2%20I%5E2%5D%7D%20%3D%20%5BI%20T%5D

.Upon evaluating each option, onlyOption D:

%5Csqrt%7B%5Cepsilon_0%20h%20c%7D

has the dimensions of an electric charge

%5BI%20T%5D

To determine the expression for

%5Calpha

in SI units, use the given relation:

T%20%3D%202%20%5Cpi%5E%7B3%2F2%7D%20%5Calpha%20%5Cfrac%7B%5Csqrt%7Bm%7D%7D%7Bq%7D

From this, the dimensional formula for

%5Calpha

is:

%5B%5Calpha%5D%20%3D%20%5Cfrac%7B%5BT%5D%5Bq%5D%7D%7B%5B%5Csqrt%7Bm%7D%5D%7D

Substituting the dimensional formulas, we have:

%5B%5Calpha%5D%20%3D%20%5Cfrac%7B%5BT%5D%5BI%20%5Ccdot%20T%5D%7D%7B%5Cleft%5BM%5E%7B1%2F2%7D%5Cright%5D%7D%20%3D%20%5Cleft%5BM%5E%7B-1%2F2%7D%20L%5E0%20T%5E2%20I%5E1%5Cright%5D

Next, consider the dimensions of the permittivity of free space

%5Cepsilon_0

, which can be derived from Coulomb's law:

F%20%3D%20%5Cfrac%7B1%7D%7B4%20%5Cpi%20%5Cepsilon_0%7D%20%5Cfrac%7Bq%5E2%7D%7Br%5E2%7D

The dimensional formula for

%5Cepsilon_0

is:

%5B%5Cepsilon_0%5D%20%3D%20%5Cleft%5BM%5E%7B-1%7D%20L%5E%7B-3%7D%20T%5E4%20I%5E2%5Cright%5D

Taking the square root of

%5Cepsilon_0

, we get:

%5B%5Cepsilon_0%5D%5E%7B1%2F2%7D%20%3D%20%5Cleft%5BM%5E%7B-1%2F2%7D%20L%5E%7B-3%2F2%7D%20T%5E2%20I%5E1%5Cright%5D

By comparing the dimensions of

%5Cepsilon_0%5E%7B1%2F2%7D

with

%5Calpha

, it is apparent that

%5Cepsilon_0%5E%7B1%2F2%7D

matches the dimensions of

%5Calpha

except for a factor of

L%5E%7B3%2F2%7D

. Thus, the expression for

%5Calpha

in terms of

%5Cepsilon_0

and

L

is:

%5Calpha%20%3D%20%5Cepsilon_0%5E%7B1%2F2%7D%20L%5E%7B3%2F2%7D

Units And Measurement And Dimensions

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Units And Measurement And Dimensions

High-Yield Trend

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3 MCQs