Select the dimensional formula of `B^2/(2mu0)`

Options

A. `[M^1L^1T^2]`
B. `[M^-1L^1T^2]`
C. `[M^-1L^-1T^-2]`
D. `[M^1L^-1T^-2]`

Answer

To find the dimensional formula of \( \frac{B^2}{2\mu_0} \), we can follow these steps:

### Step 1: Understand the components
We need to analyze the expression \( \frac{B^2}{2\mu_0} \). Here, \( B \) represents the magnetic field, and \( \mu_0 \) is the permeability of free space.

### Step 2: Find the dimensional formula of \( B \)
The magnetic field \( B \) can be related to force, charge, and velocity using the Lorentz force law:
\[ F = QvB \]
From this, we can express \( B \) as:
\[ B = \frac{F}{Qv} \]
The dimensional formula for force \( F \) is \( [F] = MLT^{-2} \), for charge \( Q \) it is \( [Q] = IT \), and for velocity \( v \) it is \( [v] = LT^{-1} \).

Substituting these into the equation for \( B \):
\[ B = \frac{MLT^{-2}}{IT \cdot LT^{-1}} = \frac{MLT^{-2}}{ILT^{-1}} = \frac{M}{I} \cdot T^{-1} \]
Thus, the dimensional formula for \( B \) is:
\[ [B] = M^1 I^{-1} T^{-1} \]

### Step 3: Find the dimensional formula of \( \mu_0 \)
The permeability of free space \( \mu_0 \) has the dimensional formula:
\[ [\mu_0] = \frac{[F]}{[B]^2} \]
Substituting the dimensional formulas we found:
\[ [\mu_0] = \frac{MLT^{-2}}{(M^1 I^{-1} T^{-1})^2} = \frac{MLT^{-2}}{M^2 I^{-2} T^{-2}} = \frac{MLT^{-2} \cdot T^{2}}{M^{2} I^{-2}} = \frac{ML}{M^2 I^{-2}} = M^{-1} L T^{2} I^{2} \]

### Step 4: Calculate the dimensional formula of \( \frac{B^2}{\mu_0} \)
Now we can calculate \( \frac{B^2}{\mu_0} \):
\[ [B^2] = (M^1 I^{-1} T^{-1})^2 = M^2 I^{-2} T^{-2} \]
Thus:
\[ \frac{B^2}{\mu_0} = \frac{M^2 I^{-2} T^{-2}}{M^{-1} L T^{2} I^{2}} = M^{2 - (-1)} I^{-2 - 2} T^{-2 - 2} L^{-1} = M^{3} I^{-4} T^{-4} L^{-1} \]

### Step 5: Final dimensional formula
The dimensional formula of \( \frac{B^2}{2\mu_0} \) is the same as that of \( \frac{B^2}{\mu_0} \) since \( 2 \) is a dimensionless constant:
\[ \frac{B^2}{2\mu_0} = M^{3} I^{-4} T^{-4} L^{-1} \]

### Conclusion
The dimensional formula of \( \frac{B^2}{2\mu_0} \) is:
\[ [M^3 I^{-4} T^{-4} L^{-1}] \]