Dimensional formula of magnetic field is :-

Options

A. `[MT^(-2)A^(-1)]`
B. `[ML^(2)T^(-1)A^(-2)]`
C. `[MT^(-2)A^(-2)]`
D. `[MT^(-1)A^(-2)]`

Answer

To find the dimensional formula of the magnetic field (B), we can start from the relationship involving force (F), charge (q), and velocity (v). The equation we will use is:

\[ F = q \cdot v \cdot B \cdot \sin(\theta) \]

From this equation, we can express the magnetic field (B) as:

\[ B = \frac{F}{q \cdot v \cdot \sin(\theta)} \]

Since \(\sin(\theta)\) is a dimensionless quantity, we can ignore it for the dimensional analysis. Now, we need to find the dimensions of force (F), charge (q), and velocity (v).

### Step 1: Determine the dimensions of force (F)
The dimensional formula for force is given by:

\[ F = m \cdot a \]

Where:
- \(m\) is mass with dimension \(M\)
- \(a\) is acceleration with dimension \(L \cdot T^{-2}\)

Thus, the dimensional formula for force is:

\[ [F] = M \cdot (L \cdot T^{-2}) = M L T^{-2} \]

### Step 2: Determine the dimensions of charge (q)
The dimensional formula for electric charge is given by:

\[ [q] = A \cdot T \]

Where:
- \(A\) is the dimension of electric current
- \(T\) is the dimension of time

Thus, the dimensional formula for charge is:

\[ [q] = A T \]

### Step 3: Determine the dimensions of velocity (v)
The dimensional formula for velocity is given by:

\[ [v] = \frac{L}{T} \]

### Step 4: Substitute the dimensions into the formula for B
Now we can substitute the dimensions of force, charge, and velocity into the formula for magnetic field B:

\[ [B] = \frac{[F]}{[q] \cdot [v]} \]

Substituting the dimensions we found:

\[ [B] = \frac{M L T^{-2}}{(A T) \cdot (L T^{-1})} \]

### Step 5: Simplify the expression
Now, simplify the expression:

\[ [B] = \frac{M L T^{-2}}{A L T^{-1}} \]

Canceling \(L\) from the numerator and denominator:

\[ [B] = \frac{M T^{-2}}{A T^{-1}} \]

Now, simplifying further:

\[ [B] = M T^{-2} \cdot A^{-1} \cdot T = M A^{-1} T^{-1} \]

### Final Result
Thus, the dimensional formula for the magnetic field (B) is:

\[ [B] = M A^{-1} T^{-1} \]