JEE Advanced · Physics · 2. Units & Dimensions

Planck' 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 the correct option(s) is (are)

A $M \propto \sqrt{c}$
B $M \propto \sqrt{G}$
C $L \propto \sqrt{h}$
D $L \propto \sqrt{G}$

Verified Solution

Answer & Solution

Correct Answer

(A) $M \propto \sqrt{c}$

Step-by-step Solution

Detailed explanation

[h] = [M L^{2} T^{- 1}]

[c] = [L T^{- 1}]

[G] = [M^{- 1} L^{3} T^{- 2}]

For unit of length
Let

L \propto h^{x} c^{y} G^{z}

\Rightarrow

From principle of homogeneity

[L H S] = [R H S]

\Rightarrow [M^{0} L T^{0}] = {[M L^{2} T^{- 1}]}^{x} {[L T^{- 1}]}^{y} {[M^{- 1} L^{3} T^{- 2}]}^{z}

M^{0} L T^{0} = M^{x - z} L^{2 x + y + 3 z} T^{- x - y - 2 z}

Comparing we get

x - z = 0

2 x + y + 3 z = 1

- x - y - 2 z = 0

Solving we get,

x = \frac{1}{2}, y = \frac{- 3}{2}, z = \frac{1}{2}

L = \sqrt{\frac{h G}{c^{3}}}

For unit of mass
Let

M \propto h^{x} c^{y} G^{z}

Solving in similar manner as above
We get

x = \frac{1}{2}, y = \frac{1}{2}, z = \frac{- 1}{2}

M = \sqrt{\frac{h c}{G}}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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List-I shows various functional dependencies of energy $(E)$ on the atomic number $(Z)$. Energies associated with certain phenomena are given in List-II. Choose the option that describes the correct match between the entries in List-I to those in List-II.

List - I	List - II
(P) $E\text { propto } Z^2$	(1) energy of characteristic x-rays
(Q) $E\text { propto } (Z-1)^2$	(2) electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range 30 to 170
(R) $E \text { propto } Z(Z-1)$	(3) energy of continuous x-rays
(S) E is practically independent of Z	(4) average nuclear binding energy per nucleon for stable nuclei with mass number in the range 30 to 170
	(5) energy of radiation due to electronic transitions from hydrogen-like atoms

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From JEE Advanced

Explore more questions on app

List - I	List - II
(P) \(E\text { propto } Z^2\)	(1) energy of characteristic x-rays
(Q) \(E\text { propto } (Z-1)^2\)	(2) electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range 30 to 170
(R) \(E \text { propto } Z(Z-1)\)	(3) energy of continuous x-rays
(S) E is practically independent of Z	(4) average nuclear binding energy per nucleon for stable nuclei with mass number in the range 30 to 170
	(5) energy of radiation due to electronic transitions from hydrogen-like atoms

Planck' 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 the correct option(s) is (are)

Step-by-step Solution

Planck' 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 the correct option(s) is (are)