Question

A dimensionless quantity is constructed in terms of electronic charge 'e', permittivity of free space [M ^ - 1 * L ^ - 3 * T ^ 4 * A ^ 3] Planck's constant h ^ 2 * {M * L ^ 2 * T ^ 1} and speed of light "c". If the dimensionless quantity is written as c ^ n * c_{0} ^ 0 * h ^ T * c ^ 0 and 'n' is a non-zero integer, what will be the values of a, B, y and & in terms of 'n

• A. (2n,-n,-n,-n)
• B. (n,-n,-2n,-n)
• C. (n,-n,-n,-2n)
• D. (2n,-n,-2n,-2n)

Answer 1

Answer: (A)

(2n,-n,-n,-n)

Answer 2

The problem asks us to find the exponents $a, B, \gamma, \delta$ for a dimensionless quantity $X = e^a \epsilon_0^B h^\gamma c^\delta$, where 'n' is a non-zero integer related to these exponents. We are given the names of the physical quantities: electronic charge 'e', permittivity of free space 'ε₀', Planck's constant 'h', and speed of light 'c'.

First, let's identify the standard dimensions of each physical quantity. It is important to note that the question contains some typos in the given dimensions for $\epsilon_0$ and $h$. We will use the standard, correct dimensions for these quantities, as is typical for such problems and consistent with the similar question provided.

1. Electronic charge 'e': The dimension of charge is current (A) times time (T). $[e] = [A T]$

2. Permittivity of free space 'ε₀': From Coulomb's law, $F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}$, so $\epsilon_0 = \frac{q_1 q_2}{4\pi F r^2}$. $[\epsilon_0] = \frac{[A T]^2}{[M L T^{-2}] [L^2]} = [M^{-1} L^{-3} T^4 A^2]$ (The question states $[M^{-1} L^{-3} T^4 A^3]$, which is incorrect. We use $A^2$.)

3. Planck's constant 'h': From the energy of a photon $E = h\nu$, where $\nu$ is frequency. $[h] = \frac{[E]}{[\nu]} = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}]$ (The question states "$h^2 \{M L^2 T^1\}$", which is incorrect. We use $h$ and $T^{-1}$.)

4. Speed of light 'c': Speed is distance over time. $[c] = [L T^{-1}]$

Let the dimensionless quantity be $X = e^a \epsilon_0^B h^\gamma c^\delta$. For $X$ to be dimensionless, its dimensions must be $[M^0 L^0 T^0 A^0]$.

Substitute the dimensions into the expression for $X$: $[X] = [A T]^a \cdot [M^{-1} L^{-3} T^4 A^2]^B \cdot [M L^2 T^{-1}]^\gamma \cdot [L T^{-1}]^\delta = [M^0 L^0 T^0 A^0]$

Now, equate the powers of M, L, T, and A on both sides:

• For Mass (M): $-B + \gamma = 0 \implies \gamma = B$ (Equation 1)

• For Length (L): $-3B + 2\gamma + \delta = 0$ (Equation 2)

• For Time (T): $a + 4B - \gamma - \delta = 0$ (Equation 3)

• For Current (A): $a + 2B = 0 \implies a = -2B$ (Equation 4)

Now, we solve this system of linear equations:

1. From Equation 1, we have $\gamma = B$.

2. Substitute $\gamma = B$ into Equation 2: $-3B + 2(B) + \delta = 0$ $-B + \delta = 0 \implies \delta = B$ (Equation 5)

3. Substitute $a = -2B$ (from Equation 4), $\gamma = B$ (from Equation 1), and $\delta = B$ (from Equation 5) into Equation 3: $(-2B) + 4B - (B) - (B) = 0$ $-2B + 4B - B - B = 0$ $2B - 2B = 0$ $0 = 0$

This result ($0=0$) indicates that the equations are consistent and that the exponents are not uniquely determined as absolute values, but rather in terms of one another. We have found the relationships: $a = -2B$ $\gamma = B$ $\delta = B$

The question asks for the values of $a, B, \gamma, \delta$ in terms of 'n', where 'n' is a non-zero integer. Comparing this with the similar question, the solution is given in the form $(2n, -n, -n, -n)$. This implies that we should set $B = -n$.

If we set $B = -n$: $a = -2(-n) = 2n$ $\gamma = -n$ $\delta = -n$

Thus, the values of the exponents are $(a, B, \gamma, \delta) = (2n, -n, -n, -n)$. This means the dimensionless quantity can be written as $(e^2 \epsilon_0^{-1} h^{-1} c^{-1})^n$. A common example of such a dimensionless quantity is the fine-structure constant, $\alpha = \frac{e^2}{4\pi\epsilon_0 \hbar c}$, which has exponents $(2, -1, -1, -1)$ for $e, \epsilon_0, h, c$ respectively (ignoring constants like $4\pi$ and $2\pi$ in $\hbar$). This corresponds to setting $n=1$ in our derived general form.

The final answer is $\boxed{(2n, -n, -n, -n)}$.