The ratio of magnetic fields on the axis of a cricular current cerrying coil of radius a to the magnetic field at its centre lying on the periphery of the surface of the sphere will be .

Options

A. a.`(1)/((1+(x^2)/(a^2))^(3//2))`
B. b.`(1)/((1+a^2/x^2)^(1//2))`
C. `c.(1)/((1+a^2/x^2)^(2))`
D. d.`(1)/((1+a^2/x^2)^(3))`

Answer

To solve the problem of finding the ratio of the magnetic field on the axis of a circular current-carrying coil of radius \( a \) to the magnetic field at its center lying on the periphery of the surface of the sphere, we will follow these steps:

### Step 1: Understand the Magnetic Field on the Axis of the Coil
The magnetic field \( B_1 \) at a distance \( x \) from the center of a circular current-carrying coil of radius \( a \) can be expressed as:
\[ B_1 = \frac{\mu_0 I a^2}{2(a^2 + x^2)^{3/2}} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current flowing through the coil,
- \( a \) is the radius of the coil,
- \( x \) is the distance from the center along the axis.

### Step 2: Calculate the Magnetic Field at the Center of the Coil
The magnetic field \( B_2 \) at the center of the coil (where \( x = 0 \)) is given by:
\[ B_2 = \frac{\mu_0 I a^2}{2a^3} = \frac{\mu_0 I}{2a} \]

### Step 3: Set Up the Ratio of the Magnetic Fields
We need to find the ratio of \( B_1 \) to \( B_2 \):
\[ \text{Ratio} = \frac{B_1}{B_2} \]

### Step 4: Substitute the Expressions for \( B_1 \) and \( B_2 \)
Substituting the expressions we derived:
\[ \text{Ratio} = \frac{\frac{\mu_0 I a^2}{2(a^2 + x^2)^{3/2}}}{\frac{\mu_0 I}{2a}} = \frac{a^2}{(a^2 + x^2)^{3/2}} \cdot a = \frac{a^3}{(a^2 + x^2)^{3/2}} \]

### Step 5: Simplify the Ratio
The ratio can be simplified further:
\[ \text{Ratio} = \frac{a^3}{(a^2 + x^2)^{3/2}} \]

### Step 6: Evaluate the Ratio for Specific Cases
If we consider the case where \( x = a \) (the distance from the center to the periphery of the sphere), we substitute \( x = a \):
\[ \text{Ratio} = \frac{a^3}{(a^2 + a^2)^{3/2}} = \frac{a^3}{(2a^2)^{3/2}} = \frac{a^3}{2^{3/2} a^3} = \frac{1}{2^{3/2}} = \frac{1}{2\sqrt{2}} \]

### Final Answer
Thus, the ratio of the magnetic field on the axis of the circular current-carrying coil to the magnetic field at its center lying on the periphery of the surface of the sphere is:
\[ \text{Ratio} = \frac{1}{2\sqrt{2}} \]

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