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 find the ratio of the magnetic fields produced by two solenoids with different lengths and number of loops, we can follow these steps:

### Step-by-Step Solution:

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

2. **Formula for Magnetic Field in 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 Magnetic Field for Solenoid A:**
For Solenoid A:
\[ B_1 = \mu_0 \frac{N}{L} I \]

4. **Calculate Magnetic Field for Solenoid B:**
For Solenoid B:
- The number of turns per unit length \( n' \) for Solenoid B is:
\[ n' = \frac{4N}{2L} = \frac{2N}{L} \]
So, the magnetic field \( B_2 \) for Solenoid B is:
\[ B_2 = \mu_0 \frac{4N}{2L} I = \mu_0 \frac{2N}{L} I \]

5. **Find the Ratio of the Magnetic Fields:**
Now, we can find the ratio of the magnetic fields \( B_1 \) and \( B_2 \):
\[ \frac{B_1}{B_2} = \frac{\mu_0 \frac{N}{L} I}{\mu_0 \frac{2N}{L} I} = \frac{N}{2N} = \frac{1}{2} \]

6. **Final Result:**
Thus, the ratio of the magnetic fields \( B_1 : B_2 \) is:
\[ B_1 : B_2 = 1 : 2 \]
Therefore, the ratio of \( B_1 \) to \( B_2 \) is \( 0.5 \).

### Conclusion:
The ratio of the magnetic fields produced by the two solenoids is \( 0.5 \).