Magnetic field near centre of long solenoid for a fixed number of turns is given by, B = KI, where K is the constant of proportionality. For 1.5 A current, the amount of magnetic field is 12 tesla, what is the value of magnetic field for the same number of turns at same point for 3 A current ?

Explanation

To solve the problem step by step, we will use the relationship between the magnetic field (B) and the current (I) in a solenoid, given by the formula \( B = KI \), where K is a constant.

### Step-by-Step Solution:

1. **Understand the relationship**: The magnetic field \( B \) near the center of a long solenoid is directly proportional to the current \( I \). This can be expressed as:
\[ B = K \cdot I \]
where \( K \) is a constant.

2. **Set up the equations for the two scenarios**:
- For the first scenario with a current \( I_1 = 1.5 \, \text{A} \), the magnetic field \( B_1 \) is given as:
\[ B_1 = K \cdot I_1 = K \cdot 1.5 \]
- For the second scenario with a current \( I_2 = 3 \, \text{A} \), the magnetic field \( B_2 \) can be expressed as:
\[ B_2 = K \cdot I_2 = K \cdot 3 \]

3. **Use the information given**: We know that for \( I_1 = 1.5 \, \text{A} \), \( B_1 = 12 \, \text{T} \). Thus, we can write:
\[ 12 = K \cdot 1.5 \]

4. **Calculate the constant \( K \)**: Rearranging the equation to find \( K \):
\[ K = \frac{12}{1.5} = 8 \, \text{T/A} \]

5. **Substitute \( K \) into the equation for \( B_2 \)**: Now that we have \( K \), we can find \( B_2 \):
\[ B_2 = K \cdot I_2 = 8 \cdot 3 = 24 \, \text{T} \]

6. **Final answer**: The value of the magnetic field for the same number of turns at the same point for a current of \( 3 \, \text{A} \) is:
\[ B_2 = 24 \, \text{T} \]

### Summary:
The magnetic field for a current of \( 3 \, \text{A} \) is \( 24 \, \text{T} \).

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