Two circular coils are made from a uniform copper wire. Radii of circular coils is in the ratio 3:4 and number of turns in the ratio 3:5. If they are connected in series across a battery.
Statement (A) : Ratio between magnetic field inductions at their centers is 4 : 5
Statement (B) : Ratio between effective magnetic moments of the two coils is 16 : 15

Options

A. Both statements are wrong
B. Both statements are correct
C. Statement A alone is correct
D. Statement B alone is correct

Answer

To solve the problem, we need to analyze the two statements regarding the magnetic field inductions and effective magnetic moments of the two circular coils.

### Given Data:
1. The radii of the circular coils are in the ratio \( R_1 : R_2 = 3 : 4 \).
2. The number of turns in the coils are in the ratio \( n_1 : n_2 = 3 : 5 \).

### Step 1: Magnetic Field Induction at the Center of the Coils
The magnetic field induction \( B \) at the center of a circular coil is given by the formula:
\[ B = \frac{\mu_0 n I}{2R} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( n \) is the number of turns,
- \( I \) is the current,
- \( R \) is the radius of the coil.

For coil 1:
\[ B_1 = \frac{\mu_0 n_1 I}{2R_1} \]

For coil 2:
\[ B_2 = \frac{\mu_0 n_2 I}{2R_2} \]

### Step 2: Ratio of Magnetic Field Inductions
Taking the ratio \( \frac{B_1}{B_2} \):
\[ \frac{B_1}{B_2} = \frac{\mu_0 n_1 I}{2R_1} \cdot \frac{2R_2}{\mu_0 n_2 I} = \frac{n_1}{n_2} \cdot \frac{R_2}{R_1} \]

Substituting the ratios:
\[ \frac{B_1}{B_2} = \frac{3/5 \cdot 4/3} = \frac{4}{5} \]

### Conclusion for Statement (A)
Thus, the ratio of magnetic field inductions at their centers is \( \frac{B_1}{B_2} = \frac{4}{5} \), which confirms that Statement (A) is **True**.

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### Step 3: Effective Magnetic Moments of the Coils
The magnetic moment \( m \) of a coil is given by:
\[ m = n I A \]
where \( A \) is the area of the coil. For a circular coil:
\[ A = \pi R^2 \]
Thus, for coil 1:
\[ m_1 = n_1 I \pi R_1^2 \]
For coil 2:
\[ m_2 = n_2 I \pi R_2^2 \]

### Step 4: Ratio of Effective Magnetic Moments
Taking the ratio \( \frac{m_1}{m_2} \):
\[ \frac{m_1}{m_2} = \frac{n_1 I \pi R_1^2}{n_2 I \pi R_2^2} = \frac{n_1}{n_2} \cdot \frac{R_1^2}{R_2^2} \]

Substituting the ratios:
\[ \frac{m_1}{m_2} = \frac{3/5 \cdot (3/4)^2} = \frac{3}{5} \cdot \frac{9}{16} = \frac{27}{80} \]

### Conclusion for Statement (B)
The ratio of effective magnetic moments is \( \frac{m_1}{m_2} = \frac{27}{80} \), which does not equal \( \frac{16}{15} \). Therefore, Statement (B) is **False**.

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### Final Conclusion
- Statement (A) is True.
- Statement (B) is False.

Thus, the answer is that only Statement (A) is correct.

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