There is one circular loop of wire of radius a, carrying current i. At what distance from its centre on its axis will the magnetic field intensity become `1/(2 sqrt2)` times at the centre?

Options

A. a
B. 2a
C. `3a//2`
D. `a//3`

Explanation

To solve the problem of finding the distance from the center of a circular loop of wire where the magnetic field intensity is \( \frac{1}{2\sqrt{2}} \) times that at the center, we can follow these steps:

### Step 1: Determine the Magnetic Field at the Center
The magnetic field \( B_0 \) at the center of a circular loop of radius \( a \) carrying current \( i \) is given by the formula:
\[ B_0 = \frac{\mu_0 i}{2a} \]
where \( \mu_0 \) is the permeability of free space.

### Step 2: Write the Expression for Magnetic Field at a Distance \( x \) on the Axis
The magnetic field \( B_x \) at a distance \( x \) from the center along the axis of the loop is given by:
\[ B_x = \frac{\mu_0 i a^2}{2(x^2 + a^2)^{3/2}} \]

### Step 3: Set Up the Equation for \( B_x \)
According to the problem, we want to find the distance \( x \) where the magnetic field \( B_x \) is \( \frac{1}{2\sqrt{2}} B_0 \):
\[ B_x = \frac{1}{2\sqrt{2}} B_0 \]
Substituting the expression for \( B_0 \):
\[ B_x = \frac{1}{2\sqrt{2}} \left( \frac{\mu_0 i}{2a} \right) \]

### Step 4: Equate the Two Expressions
Now we equate the two expressions for \( B_x \):
\[ \frac{\mu_0 i a^2}{2(x^2 + a^2)^{3/2}} = \frac{1}{2\sqrt{2}} \left( \frac{\mu_0 i}{2a} \right) \]
We can cancel \( \frac{\mu_0 i}{2} \) from both sides:
\[ \frac{a^2}{(x^2 + a^2)^{3/2}} = \frac{1}{2\sqrt{2} a} \]

### Step 5: Cross-Multiply to Solve for \( x \)
Cross-multiplying gives:
\[ a^2 \cdot 2\sqrt{2} a = (x^2 + a^2)^{3/2} \]
This simplifies to:
\[ 2\sqrt{2} a^3 = (x^2 + a^2)^{3/2} \]

### Step 6: Raise Both Sides to the Power of \( \frac{2}{3} \)
To eliminate the exponent on the right side, we raise both sides to the power of \( \frac{2}{3} \):
\[ (2\sqrt{2} a^3)^{\frac{2}{3}} = x^2 + a^2 \]

### Step 7: Simplify and Solve for \( x^2 \)
Calculating the left side:
\[ (2\sqrt{2})^{\frac{2}{3}} a^2 = x^2 + a^2 \]
This gives:
\[ \frac{4}{\sqrt{2}} a^2 = x^2 + a^2 \]
Subtract \( a^2 \) from both sides:
\[ \frac{4}{\sqrt{2}} a^2 - a^2 = x^2 \]
\[ \left( \frac{4}{\sqrt{2}} - 1 \right) a^2 = x^2 \]

### Step 8: Calculate \( x \)
Now, we can calculate \( x \):
\[ x = \sqrt{\left( \frac{4}{\sqrt{2}} - 1 \right) a^2} = a \sqrt{\left( \frac{4}{\sqrt{2}} - 1 \right)} \]

### Step 9: Final Result
After simplifying, we find:
\[ x = a \sqrt{2} - a = a(\sqrt{2} - 1) \]