Two solenoids having lengths L and 2L and the number of loops N and 4N, both have the same current, then ihe ratio of the magnetic field will be

Options

A. 1
B. 0.5
C. 2
D. 0.25

Explanation

To solve the problem, we need to find the ratio of the magnetic fields produced by two solenoids with different lengths and number of turns.

### Step-by-Step Solution:

1. **Identify the Parameters of the Solenoids:**
- For Solenoid A:
- Length (L) = L
- Number of turns (N) = N
- Current (I) = I
- For Solenoid B:
- Length (L) = 2L
- Number of turns (N) = 4N
- Current (I) = I

2. **Formula for Magnetic Field inside a Solenoid:**
The magnetic field \( B \) inside a solenoid is given by the formula:
\[ B = \mu_0 \frac{N}{L} I \]
where \( \mu_0 \) is the permeability of free space, \( N \) is the number of turns, \( L \) is the length of the solenoid, and \( I \) is the current.

3. **Calculate the Magnetic Field for Solenoid A (B1):**
- For Solenoid A:
\[ B_1 = \mu_0 \frac{N}{L} I \]

4. **Calculate the Magnetic Field for Solenoid B (B2):**
- For Solenoid B:
\[ B_2 = \mu_0 \frac{4N}{2L} I \]
Simplifying this:
\[ B_2 = \mu_0 \frac{4N}{2L} I = \mu_0 \frac{2N}{L} I \]

5. **Express B2 in terms of B1:**
- From the expressions derived:
\[ B_1 = \mu_0 \frac{N}{L} I \quad \text{and} \quad B_2 = \mu_0 \frac{2N}{L} I \]
We can see that:
\[ B_2 = 2 B_1 \]

6. **Calculate the Ratio of the Magnetic Fields:**
- The ratio of the magnetic fields \( \frac{B_1}{B_2} \) is:
\[ \frac{B_1}{B_2} = \frac{B_1}{2B_1} = \frac{1}{2} \]

7. **Final Answer:**
- Therefore, the ratio of the magnetic fields \( B_1 : B_2 \) is:
\[ B_1 : B_2 = 1 : 2 \]

### Conclusion:
The ratio of the magnetic fields produced by the two solenoids is \( 1 : 2 \).