Given a charge q$q$, current I$I$ and permeability of vacuum mu_0$\mu_0$ . Which of the following quantity has the dimension of momentum?

### Related Formula Let momentum be P$P$ ([P] = [M L T^-1]$[P] = [M L T^{-1}]$). We assume: P = q^x mu_0^y I^z$$P = q^x \mu_0^y I^z$$ ### Core Logic Let's find the dimensional formulas of the individual variables: 1. **Charge (q$q$):** [q] = [A T]$[q] = [A T]$ 2. **Current (I$I$):** [I] = [A]$[I] = [A]$ 3. **Permeability of vacuum (mu_0$\mu_0$):** From Biot-Savart law or force between parallel wires: F = fracmu_0 I^2 L2pi d$F = \frac{\mu_0 I^2 L}{2\pi d}$: [mu_0] = frac[F][I]^2 = frac[M L T^-2][A]^2 = [M L T^-2 A^-2]$$[\mu_0] = \frac{[F]}{[I]^2} = \frac{[M L T^{-2}]}{[A]^2} = [M L T^{-2} A^{-2}]$$ ### Step 1: Apply Dimensional Homogeneity Substitute these into our assumed dimensional equation: [M L T^-1] = [A T]^x [M L T^-2 A^-2]^y [A]^z$$[M L T^{-1}] = [A T]^x [M L T^{-2} A^{-2}]^y [A]^z$$ [M L T^-1] = [M]^y [L]^y [T]^x - 2y [A]^x - 2y + z$$[M L T^{-1}] = [M]^y [L]^y [T]^{x - 2y} [A]^{x - 2y + z}$$ Comparing exponents on both sides: - For [M]$[M]$: y = 1$y = 1$ - For [L]$[L]$: y = 1 quad text(consistent)$$y = 1 \quad \text{(consistent)}$$ - For [T]$[T]$: x - 2y = -1 implies x - 2(1) = -1 implies x = 1$$x - 2y = -1 \implies x - 2(1) = -1 \implies x = 1$$ - For [A]$[A]$: x - 2y + z = 0 implies 1 - 2(1) + z = 0 implies z = 1$$x - 2y + z = 0 \implies 1 - 2(1) + z = 0 \implies z = 1$$ Thus, x = 1, y = 1, z = 1$x = 1, y = 1, z = 1$. Therefore, the required quantity is: q^1 mu_0^1 I^1 = q mu_0 I$$q^1 \mu_0^1 I^1 = q \mu_0 I$$ ### Pattern Recognition Sees: Permeability, charge, and current linked to momentum. Trap: Deriving the dimensions of mu_0$\mu_0$ using complex magnetic formulas. Remember [mu_0] = [textForce]/[textCurrent]^2$[\mu_0] = [\text{Force}]/[\text{Current}]^2$ is the quickest way to get its dimensions. Shortcut: Since [q] = AT$[q] = AT$ and [I] = A$[I] = A$, [q mu_0 I] = [A T] [M L T^-2 A^-2] [A] = [M L T^-1]$[q \mu_0 I] = [A T] [M L T^{-2} A^{-2}] [A] = [M L T^{-1}]$, which is exactly the dimensions of momentum. Hence, option (2) is correct. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements Class 12 Physics: Moving Charges and Magnetism