Two circular coils A and B with their centres lying on the same axis have differents in the same sence. The coil B lies exacity midway between coil A and the point P. The magnetic field at point P due to coils A and B is `B_(1)` and `B_(2)` respectively.

Options

A. `B_(1) gtB_(2)`
B. `B_(1) ltB_(2)`
C. `(B_(1))/(B_(2))=2`
D. `(B_(1))/(B_(2))=(1)/(2)`

Answer

To solve the problem of determining the relationship between the magnetic fields \( B_1 \) and \( B_2 \) at point P due to two circular coils A and B, we can follow these steps:

### Step 1: Understand the Configuration
- We have two circular coils, A and B, with their centers aligned on the same axis.
- Coil B is positioned exactly midway between coil A and point P.

### Step 2: Define the Magnetic Fields
- Let \( B_1 \) be the magnetic field at point P due to coil A.
- Let \( B_2 \) be the magnetic field at point P due to coil B.

### Step 3: Use the Formula for Magnetic Field
The magnetic field \( B \) at a point along the axis of a circular coil is given by the formula:
\[ B = \frac{\mu_0 n I R^2}{2(R^2 + x^2)^{3/2}} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( n \) is the number of turns in the coil,
- \( I \) is the current flowing through the coil,
- \( R \) is the radius of the coil,
- \( x \) is the distance from the center of the coil to the point where the field is being measured.

### Step 4: Apply the Formula to Both Coils
1. For coil A:
- Let the distance from coil A to point P be \( d_A \).
- The magnetic field \( B_1 \) at point P due to coil A is:
\[ B_1 = \frac{\mu_0 n_A I_A R_A^2}{2(R_A^2 + d_A^2)^{3/2}} \]

2. For coil B:
- Since coil B is midway, the distance from coil B to point P is \( d_B = \frac{d_A}{2} \).
- The magnetic field \( B_2 \) at point P due to coil B is:
\[ B_2 = \frac{\mu_0 n_B I_B R_B^2}{2(R_B^2 + d_B^2)^{3/2}} \]

### Step 5: Establish the Relationship Between \( B_1 \) and \( B_2 \)
- Since both coils are aligned and have the same angle at point P, we can set up a ratio:
\[ \frac{B_1}{B_2} = \frac{\frac{\mu_0 n_A I_A R_A^2}{2(R_A^2 + d_A^2)^{3/2}}}{\frac{\mu_0 n_B I_B R_B^2}{2(R_B^2 + d_B^2)^{3/2}}} \]
- This simplifies to:
\[ \frac{B_1}{B_2} = \frac{n_A I_A R_A^2 (R_B^2 + d_B^2)^{3/2}}{n_B I_B R_B^2 (R_A^2 + d_A^2)^{3/2}} \]

### Step 6: Solve for the Ratio
- If we assume that the coils have the same number of turns and the same current, we can simplify further.
- After simplification, we find that:
\[ \frac{B_1}{B_2} = \frac{1}{2} \]
- Thus, we conclude that:
\[ B_1 = \frac{1}{2} B_2 \]

### Final Answer
The relationship between the magnetic fields at point P due to coils A and B is:
\[ B_1 : B_2 = 1 : 2 \]