Question

A circular coil, carrying current, has radius \( R \). The distance from the centre of the coil on the axis where the magnetic induction will be \( \frac{1}{27}\text{th} \) of its value at the centre of the coil is

• A. \( 2\sqrt{2} R \)
• B. \( 3\sqrt{2} R \)
• C. \( 3 R \)
• D. \( 2\sqrt{3} R \)

Hint:

To solve questions involving \( B(x) = \frac{1}{n} B_0 \) quickly, use the direct relation:
\[ \left( 1 + \frac{x^2}{R^2} \right) = n^{2/3} \]
For \( n = 27 \), we get \( n^{2/3} = (27)^{2/3} = 9 \). Thus, \( \frac{x^2}{R^2} = 9 - 1 = 8 \implies x = \sqrt{8}R = 2\sqrt{2}R \). This shortcut saves valuable calculation time.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A circular current loop produces a magnetic field. The strength of this field is maximum at the center of the loop and decreases as we move along the axis passing through the center.
The formula for the axial magnetic field involves the radius of the loop and the distance from the center.
The problem asks for the specific distance \(x\) where the field strength drops to a specific fraction (\(1/27\)) of the field strength at the center (\(B_c\)).
Step 2: Key Formula or Approach:
Magnetic field at a distance \(x\) on the axis: \(B_x = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}\).
Magnetic field at the center (\(x = 0\)): \(B_c = \frac{\mu_0 I}{2R}\).
Condition: \(B_x = \frac{1}{27} B_c\).
Step 3: Detailed Explanation:
Substitute the given condition into the formulas:
\[ \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} = \frac{1}{27} \times \frac{\mu_0 I}{2R} \]
Cancel the common constants \(\mu_0\), \(I\), and \(2\) from both sides of the equation:
\[ \frac{R^2}{(R^2 + x^2)^{3/2}} = \frac{1}{27R} \]
Cross-multiply to simplify:
\[ 27 R^3 = (R^2 + x^2)^{3/2} \]
To solve for \(x\), take the \(2/3\) power of both sides:
\[ (27 R^3)^{2/3} = ((R^2 + x^2)^{3/2})^{2/3} \]
\[ (27)^{2/3} (R^3)^{2/3} = R^2 + x^2 \]
Calculate \(27^{2/3}\):
\[ 27^{2/3} = (3^3)^{2/3} = 3^2 = 9 \]
So the equation becomes:
\[ 9 R^2 = R^2 + x^2 \]
\[ 8 R^2 = x^2 \]
Taking the square root on both sides:
\[ x = \sqrt{8 R^2} \]
\[ x = 2\sqrt{2} R \]
Thus, the distance from the center is \(2\sqrt{2} R\), which corresponds to option (A).
Step 4: Final Answer:
By setting up the ratio of the magnetic field at a point on the axis to the field at the center, the distance \(x\) is found to be \(2\sqrt{2} R\).