If the magnetic induction at the centre of a circular coil carrying current I is 5SQRT5 times the magnetic induction at a point on its axis. Find the distance of the point on the axis from the centre of the coil. The radius of the coil is 10 cm.

To solve the problem of finding the distance from the center of a circular coil to a point on its axis where the magnetic induction is related to the magnetic induction at the center, we can use the formulas for magnetic induction due to a circular coil. Let's break this down step by step.

Understanding Magnetic Induction

Magnetic induction (or magnetic field strength) at the center of a circular coil carrying current can be calculated using the formula:

B_center = (μ₀ * I) / (2 * R)

Where:

• B_center is the magnetic induction at the center of the coil.
• μ₀ is the permeability of free space (approximately 4π x 10^-7 T·m/A).
• I is the current flowing through the coil.
• R is the radius of the coil.

Magnetic Induction on the Axis

The magnetic induction at a point along the axis of the coil at a distance x from the center is given by:

B_axis = (μ₀ * I * R²) / (2 * (R² + x²)^(3/2))

Setting Up the Equation

According to the problem, the magnetic induction at the center is 5√5 times the magnetic induction at the point on the axis. Therefore, we can set up the equation:

B_center = 5√5 * B_axis

Substituting the Formulas

Now, substituting the expressions for B_center and B_axis into the equation gives:

(μ₀ * I) / (2 * R) = 5√5 * (μ₀ * I * R²) / (2 * (R² + x²)^(3/2))

Simplifying the Equation

We can cancel out common terms (μ₀ and I) from both sides:

1 / (2 * R) = 5√5 * (R²) / (2 * (R² + x²)^(3/2))

Now, simplifying further, we get:

1 = 5√5 * R² / (R² + x²)^(3/2)

Cross-Multiplying and Rearranging

Cross-multiplying gives:

(R² + x²)^(3/2) = 5√5 * R²

Next, we can cube both sides to eliminate the exponent:

R² + x² = (5√5 * R²)^(2/3)

Substituting the Radius

Given that the radius R is 10 cm (or 0.1 m), we can substitute this value into the equation:

0.1² + x² = (5√5 * 0.1²)^(2/3)

Calculating this gives:

0.01 + x² = (5√5 * 0.01)^(2/3)

Now, calculate the right side:

5√5 * 0.01 = 0.1118

Then, raise it to the power of 2/3:

(0.1118)^(2/3) ≈ 0.046

Final Steps to Solve for x

Now we have:

0.01 + x² = 0.046

Subtracting 0.01 from both sides gives:

x² = 0.046 - 0.01 = 0.036

Taking the square root of both sides results in:

x ≈ 0.6 m

Conclusion

The distance from the center of the coil to the point on its axis where the magnetic induction is 5√5 times that at the center is approximately 0.6 meters, or 60 cm.