The dimensional formula of magnetic induction B is

Options

A. `[M^0ALT^0]`
B. `[M^0AL^(-1)T^0]`
C. `[M^0AL^2T^0]`
D. `[M^0A^(-1)T^(-2)]`

Answer

To find the dimensional formula of magnetic induction \( B \), we can start from the relationship between force, current, length, and magnetic induction. Here’s a step-by-step solution:

### Step 1: Understand the relationship
The force \( F \) on a current-carrying wire in a magnetic field is given by the equation:
\[ F = I \cdot L \cdot B \]
where:
- \( F \) is the force,
- \( I \) is the current,
- \( L \) is the length of the wire,
- \( B \) is the magnetic induction.

### Step 2: Write the dimensional formula for force
The dimensional formula for force \( F \) can be expressed as:
\[ F = m \cdot a \]
where \( m \) is mass and \( a \) is acceleration. The dimensional formula for acceleration is \( L \cdot T^{-2} \). Therefore, the dimensional formula for force is:
\[ [F] = M \cdot L \cdot T^{-2} \]

### Step 3: Write the dimensional formula for current and length
The dimensional formula for current \( I \) is:
\[ [I] = A \quad (\text{Ampere}) \]
The dimensional formula for length \( L \) is:
\[ [L] = L \]

### Step 4: Rearranging the equation for \( B \)
From the equation \( F = I \cdot L \cdot B \), we can rearrange it to find \( B \):
\[ B = \frac{F}{I \cdot L} \]

### Step 5: Substitute the dimensional formulas
Now, substituting the dimensional formulas into the equation:
\[ [B] = \frac{[F]}{[I] \cdot [L]} = \frac{M \cdot L \cdot T^{-2}}{A \cdot L} \]

### Step 6: Simplify the expression
When we simplify the expression, we can cancel \( L \) in the numerator and denominator:
\[ [B] = \frac{M \cdot L \cdot T^{-2}}{A \cdot L} = \frac{M \cdot T^{-2}}{A} = M \cdot L^{0} \cdot A^{-1} \cdot T^{-2} \]

### Final Answer
Thus, the dimensional formula of magnetic induction \( B \) is:
\[ [B] = M \cdot A^{-1} \cdot T^{-2} \]