If the strength of the magnetic field is increased by `21%` the frequency of a magnetic needle oscillating in the field.

Options

A. increased by `10%`
B. Decreases by `10%`
C. Increases by `11%`
D. Decreased by `21%`

Explanation

To solve the problem of how the frequency of a magnetic needle oscillating in a magnetic field changes when the strength of the magnetic field is increased by 21%, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship between Frequency and Magnetic Field**:
The frequency (f) of a magnetic needle oscillating in a magnetic field is related to the magnetic field strength (B). The formula for frequency is given by:
\[ f = \frac{1}{2\pi} \sqrt{\frac{m}{I}} \sqrt{B} \]
where \( m \) is the mass of the needle, \( I \) is the moment of inertia, and \( B \) is the magnetic field strength.

2. **Identifying the Change in Magnetic Field**:
The problem states that the magnetic field strength is increased by 21%. If we denote the initial magnetic field strength as \( B_1 \), then the new magnetic field strength \( B_2 \) can be expressed as:
\[ B_2 = B_1 + 0.21 B_1 = 1.21 B_1 \]

3. **Finding the Ratio of Frequencies**:
To find the change in frequency, we can take the ratio of the new frequency \( f_2 \) to the initial frequency \( f_1 \):
\[ \frac{f_2}{f_1} = \frac{\sqrt{B_2}}{\sqrt{B_1}} = \frac{\sqrt{1.21 B_1}}{\sqrt{B_1}} = \sqrt{1.21} \]

4. **Calculating the Square Root**:
We can calculate \( \sqrt{1.21} \):
\[ \sqrt{1.21} = 1.1 \]
Therefore, we have:
\[ \frac{f_2}{f_1} = 1.1 \]

5. **Finding the Percentage Change in Frequency**:
The percentage change in frequency can be calculated as:
\[ \text{Percentage Change} = \left(\frac{f_2 - f_1}{f_1}\right) \times 100 = \left(1.1 - 1\right) \times 100 = 0.1 \times 100 = 10\% \]

### Final Answer:
The frequency of the magnetic needle oscillating in the field has increased by **10%**.

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