The radius of la circular current carrying coil is R. At what distance from the centre of the coil on its axis, the intensity of magnetic field will be `1/(2sqrt(2))` times that at the centre?

Options

A. `2R`
B. `(3R)/2`
C. R is doubled
D. `R/2`

Answer

To solve the problem, we need to find the distance \( x \) from the center of a circular current-carrying coil along its axis, where the intensity of the magnetic field \( B_x \) is \( \frac{1}{2\sqrt{2}} \) times the magnetic field intensity at the center \( B_c \).

### Step 1: Understand the Magnetic Field at the Center and on the Axis
The magnetic field at the center of a circular current-carrying coil of radius \( R \) carrying current \( I \) is given by the formula:
\[ B_c = \frac{\mu_0 I}{2R} \]
where \( \mu_0 \) is the permeability of free space.

The magnetic field at a distance \( x \) along the axis of the coil is given by:
\[ B_x = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} \]

### Step 2: Set Up the Equation
We are given that:
\[ B_x = \frac{1}{2\sqrt{2}} B_c \]
Substituting the expressions for \( B_x \) and \( B_c \):
\[ \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} = \frac{1}{2\sqrt{2}} \left(\frac{\mu_0 I}{2R}\right) \]

### Step 3: Simplify the Equation
Cancel out \( \mu_0 I \) from both sides:
\[ \frac{R^2}{2(R^2 + x^2)^{3/2}} = \frac{1}{4\sqrt{2}R} \]
Now, multiply both sides by \( 4\sqrt{2}R \):
\[ 4\sqrt{2}R \cdot \frac{R^2}{2(R^2 + x^2)^{3/2}} = 1 \]
This simplifies to:
\[ 2\sqrt{2}R^3 = (R^2 + x^2)^{3/2} \]

### Step 4: Isolate \( R^2 + x^2 \)
Now, we can cube both sides to eliminate the exponent:
\[ (2\sqrt{2}R^3)^2 = R^2 + x^2 \]
Calculating the left side:
\[ 8R^6 = R^2 + x^2 \]

### Step 5: Rearranging the Equation
Rearranging gives us:
\[ x^2 = 8R^6 - R^2 \]

### Step 6: Solve for \( x \)
Taking the square root of both sides:
\[ x = \sqrt{8R^6 - R^2} \]

### Step 7: Factor Out \( R^2 \)
Factoring out \( R^2 \):
\[ x = R \sqrt{8R^4 - 1} \]

### Step 8: Find the Distance
To find the specific distance where \( B_x \) is \( \frac{1}{2\sqrt{2}} B_c \), we can substitute \( R \) into the equation:
\[ x = R \sqrt{8R^4 - 1} \]

### Final Answer
Thus, the distance from the center of the coil on its axis where the intensity of the magnetic field is \( \frac{1}{2\sqrt{2}} \) times that at the center is:
\[ x = R \]

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