Which of the following is the dimensional formula of Lorentz Force ?

Options

A. `MLT ^(-2) A^(-1)`
B. `MLT ^(-2) A ^(-1)`
C. `ML ^(2) T ^(-3) A ^(-2)`
D. `MLT ^(-2)`

Explanation

To find the dimensional formula of the Lorentz force, we can follow these steps:

### Step 1: Understand the Lorentz Force Formula
The Lorentz force \( F \) on a charged particle is given by the equation:
\[ F = qE + q(\mathbf{v} \times \mathbf{B}) \]
where:
- \( F \) is the Lorentz force,
- \( q \) is the charge,
- \( E \) is the electric field,
- \( \mathbf{v} \) is the velocity of the charged particle,
- \( \mathbf{B} \) is the magnetic field.

### Step 2: Identify the Dimensions of Each Component
1. **Charge (\( q \))**: The dimensional formula for electric charge is given as \([I \cdot T]\), where \( I \) is current and \( T \) is time.
2. **Electric Field (\( E \))**: The electric field is defined as force per unit charge:
\[ E = \frac{F}{q} \]
The dimensional formula for force \( F \) is \([M L T^{-2}]\). Thus, the dimensional formula for electric field \( E \) becomes:
\[ [E] = \frac{[F]}{[q]} = \frac{[M L T^{-2}]}{[I \cdot T]} = [M L T^{-3} I^{-1}] \]

3. **Magnetic Field (\( B \))**: The magnetic field is defined as force per unit charge and velocity:
\[ B = \frac{F}{qv} \]
Thus, the dimensional formula for magnetic field \( B \) is:
\[ [B] = \frac{[F]}{[q][v]} = \frac{[M L T^{-2}]}{[I \cdot T][L T^{-1}]} = [M T^{-2} I^{-1}] \]

### Step 3: Calculate the Dimensions of the Lorentz Force
Now, substituting the dimensions of \( E \) and \( B \) back into the Lorentz force equation:
- The term \( qE \):
\[ [qE] = [q][E] = [I \cdot T][M L T^{-3} I^{-1}] = [M L T^{-2}] \]
- The term \( q(\mathbf{v} \times \mathbf{B}) \):
\[ [q(\mathbf{v} \times \mathbf{B})] = [q][v][B] = [I \cdot T][L T^{-1}][M T^{-2} I^{-1}] = [M L T^{-2}] \]

### Step 4: Conclusion
Since both terms \( qE \) and \( q(\mathbf{v} \times \mathbf{B}) \) have the same dimensional formula as force, we conclude that the dimensional formula of the Lorentz force is:
\[ [F] = [M L T^{-2}] \]

### Final Answer
Thus, the dimensional formula of the Lorentz force is:
\[ \text{Dimensional Formula of Lorentz Force} = [M L T^{-2}] \]