Question

The fractional change in the magnetic field intensity at a distance 'r' from centre on the axis of current carrying coil of radius 'a' to the magnetic field intensity at the centre of the same coil is : (Take $r \ll a$)

• A. $\frac{3}{2} \frac{a^2}{r^2}$
• B. $\frac{2}{3} \frac{a^2}{r^2}$
• C. $\frac{2}{3} \frac{r^2}{a^2}$
• D. $\frac{3}{2} \frac{r^2}{a^2}$

Hint:

Binomial expansions $(1+x)^n \approx 1+nx$ are very common in "fractional change" problems where one variable is much smaller than the other.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The magnetic field due to a circular coil at an axial point is compared with the field at the center. For points close to the center, we use a binomial approximation.
Step 2: Key Formula or Approach:
Magnetic field at axial distance \(r\): \(B = \frac{\mu_0 I a^2}{2(a^2 + r^2)^{3/2}}\).
Field at center (\(r=0\)): \(B_0 = \frac{\mu_0 I}{2a}\).
Step 3: Detailed Explanation:
Express \(B\) in terms of \(B_0\):
\[ B = B_0 \cdot \frac{a^3}{(a^2 + r^2)^{3/2}} = B_0 \left( 1 + \frac{r^2}{a^2} \right)^{-3/2} \]
Since \(r \ll a\), apply binomial expansion \((1+x)^n \approx 1 + nx\):
\[ B \approx B_0 \left( 1 - \frac{3}{2} \frac{r^2}{a^2} \right) \]
Fractional change:
\[ \frac{B_0 - B}{B_0} = 1 - \left( 1 - \frac{3}{2} \frac{r^2}{a^2} \right) = \frac{3}{2} \frac{r^2}{a^2} \]
Step 4: Final Answer:
The fractional change is $\frac{3}{2} \frac{r^2}{a^2}$.