Understanding Planck Length Derivation
The Planck length ($L_P$) represents the scale at which the effects of quantum gravity become significant. It can be derived using dimensional analysis from the fundamental constants: the gravitational constant ($G$), the reduced Planck constant ($\hbar$), and the speed of light ($c$).
Dimensional Analysis Setup
We need to find the combination of $G$, $\hbar$, and $c$ that results in dimensions of length [L]. Let's list the dimensions of each constant:
- $G$: $[M^{-1} L^3 T^{-2}]$
- $\hbar$: $[M L^2 T^{-1}]$
- $c$: $[L T^{-1}]$
Assume the Planck length is given by $L_P = G^a \hbar^b c^d$. We equate the dimensions:
$[L] = [M^{-1} L^3 T^{-2}]^a [M L^2 T^{-1}]^b [L T^{-1}]^d$
$[L^1 M^0 T^0] = [M^{-a+b} L^{3a+2b+d} T^{-2a-b-d}]$
Solving for Exponents
Equating the exponents for each dimension (Mass M, Length L, Time T):
- Mass (M): $-a + b = 0 \implies a = b$
- Time (T): $-2a - b - d = 0$. Substituting $a=b$, we get $-2a - a - d = 0 \implies -3a - d = 0 \implies d = -3a$
- Length (L): $3a + 2b + d = 1$. Substituting $a=b$ and $d=-3a$, we get $3a + 2a + (-3a) = 1 \implies 5a - 3a = 1 \implies 2a = 1 \implies a = \frac{1}{2}$
From these results:
- $a = \frac{1}{2}$
- $b = a = \frac{1}{2}$
- $d = -3a = -\frac{3}{2}$
Final Planck Length Formula
Substituting the exponents back into the formula $L_P = G^a \hbar^b c^d$:
$L_P = G^{1/2} \hbar^{1/2} c^{-3/2}$
This can be rewritten as:
$L_P = \sqrt{\frac{G \hbar}{c^3}} = \left(\frac{G\hbar}{c^3}\right)^{1/2}$
This matches Option 3.
Similar Questions
A quantity $Q$ is formulated as $X^{-2} Y^{\frac{3}{2}} Z^{\frac{-2}{5}}$. $X, Y$ and $Z$ are independent parameters which have fractional errors of $0.1, 0.2$ and $0.5$, respectively in measurement. The maximum fractional error of $Q$ is
A man in a car at location Q on a straight highway is moving with speed v. He decides to reach a point P in qa field at a distance d from the highway (point m) as shown in the figure. Speed of the car in the field is half to that on the highway. What should be the distance RM, so that the time taken to reach P is minimum ?
- Given a charge $q$, current $I$ and permeability of vacuum $\mu_o$. Which of the following quantity has the dimension of momentum ?
- Three particles P, Q and R are moving along the vectors $\vec{A} = \hat{i} + \hat{j}$, $\vec{B} = \hat{j} + \hat{k}$ and $\vec{C} = -\hat{i} + \hat{j}$ respectively. They strike on a point and start to move in different directions.<br>Now particle P is moving normal to the plane which contains vector $\vec{A}$ and $\vec{B}$. Similarly particle Q is moving normal to the plane which contains vector $\vec{A}$ and $\vec{C}$. The angle between the direction of motion of P and Q is $\cos^{-1}\left(\frac{1}{\sqrt{x}}\right)$. Then the value of x is __________.
Three students $S_1$, $S_2$ and $S_3$ perform an experiment for determining the acceleration due to gravity (g) using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.
(least count of length =0.1 cm
least count for time =0.1s )
If $E_1$, $E_2$ and $E_3$ are the percentage errors in 'g' for students 1, 2 and 3 respectively, then the minimum percentage error is obtained by student no. __________.
- Keeping the significant figures in view, the sum of the physical quantities $52.01\text{ m}$, $153.2\text{ m}$ and $0.123\text{ m}$ is :
- When the position vector $\overrightarrow{r} = x\hat{i} + y\hat{j} + z\hat{k}$ changes sign as $-\overrightarrow{r}$, which one of the following vector will not flip under sign change ?
- For three unit vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $|\vec{a}-\vec{b}|^2 + |\vec{b}-\vec{c}|^2 + |\vec{c}-\vec{a}|^2 = 9$ and $|2\vec{a}+k\vec{b}+k\vec{c}| = 3$, the positive value of k is
- Let PQR be a triangle such that $\vec{PQ} = -2\hat{i} - \hat{j} + 2\hat{k}$ and $\vec{PR} = a\hat{i} + b\hat{j} - 4\hat{k}$, $a, b \in \mathbb{Z}$. Let S be the point on QR, which is equidistant from the lines PQ and PR. If $|\vec{PR}| = 9$ and $\vec{PS} = \hat{i} - 7\hat{j} + 2\hat{k}$, then the value of $3a - 4b$ is ________
The diameter of a wire measured by a screw gauge of least count $0.001\text{ cm}$ is $0.08\text{ cm}$. The length measured by a scale of least count $0.1\text{ cm}$ is $150\text{ cm}$. When a weight of $100\text{ N}$ is applied to the wire, the extension in length is $0.5\text{ cm}$, measured by a micrometer of least count $0.001\text{ cm}$. The error in the measured Young's modulus is $\alpha \times 10^9\text{ N/m}^2$. The value of $\alpha$ is ________.
(Ignore the contribution of the load to Young's modulus error calculation)
Important Questions from General Physics
A quantity $Q$ is formulated as $X^{-2} Y^{\frac{3}{2}} Z^{\frac{-2}{5}}$. $X, Y$ and $Z$ are independent parameters which have fractional errors of $0.1, 0.2$ and $0.5$, respectively in measurement. The maximum fractional error of $Q$ is
A man in a car at location Q on a straight highway is moving with speed v. He decides to reach a point P in qa field at a distance d from the highway (point m) as shown in the figure. Speed of the car in the field is half to that on the highway. What should be the distance RM, so that the time taken to reach P is minimum ?
- Given a charge $q$, current $I$ and permeability of vacuum $\mu_o$. Which of the following quantity has the dimension of momentum ?
- Three particles P, Q and R are moving along the vectors $\vec{A} = \hat{i} + \hat{j}$, $\vec{B} = \hat{j} + \hat{k}$ and $\vec{C} = -\hat{i} + \hat{j}$ respectively. They strike on a point and start to move in different directions.<br>Now particle P is moving normal to the plane which contains vector $\vec{A}$ and $\vec{B}$. Similarly particle Q is moving normal to the plane which contains vector $\vec{A}$ and $\vec{C}$. The angle between the direction of motion of P and Q is $\cos^{-1}\left(\frac{1}{\sqrt{x}}\right)$. Then the value of x is __________.
Three students $S_1$, $S_2$ and $S_3$ perform an experiment for determining the acceleration due to gravity (g) using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.
(least count of length =0.1 cm
least count for time =0.1s )
If $E_1$, $E_2$ and $E_3$ are the percentage errors in 'g' for students 1, 2 and 3 respectively, then the minimum percentage error is obtained by student no. __________.
