Physics Q38 – JEE-MAIN 2025

Home Physics Moving ChargesQuestion #355

Let

B_{1}

be the magnitude of magnetic field at center of a circular coil of radius

R

carrying current

I

. Let

B_{2}

be the magnitude of magnetic field at an axial distance '

x

' from the center. For

x : R = 3 : 4

\frac{B_{2}}{B_{1}}

is :

Clue 1 of 3: Recall Magnetic Field Formulas

Remember the standard formulas for the magnetic field at the center and along the axis of a circular current-carrying coil.

🥷

Ninja Strategy•Physical Intuition and Power Analysis

First, eliminate options where $B_{2} / B_{1} > 1$ . Then, recognize that the formula involves a $3 / 2$ power, suggesting a cubic relationship in the final ratio, which points to option 4.

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● Video Walkthrough

Step 1: State the Magnetic Field Formulas✦ Active

The magnetic field at the center of a circular coil of radius $R$ carrying current $I$ is given by:

B_{1} = \frac{μ_{0} I}{2 R}

The magnetic field at an axial distance $x$ from the center of the same coil is given by:

B_{2} = \frac{μ_{0} I R^{2}}{2 (R^{2} + x^{2})^{3 / 2}}

Step 2: Calculate the Ratio

B_{2} / B_{1}

○ Expand

Divide the expression for $B_{2}$ by $B_{1}$ to find the ratio:

\frac{B_{2}}{B_{1}} = \frac{\frac{μ_{0} I R^{2}}{2 (R^{2} + x^{2})^{3 / 2}}}{\frac{μ_{0} I}{2 R}} = \frac{R^{3}}{(R^{2} + x^{2})^{3 / 2}}

💡 Teacher's Secret Hint

Ensure careful cancellation of common terms like $μ_{0} I$ and $2$ .

Step 3: Substitute the Given Ratio and Simplify○ Expand

Given $x : R = 3 : 4$ , which implies $x = \frac{3}{4} R$ . Substitute this into the ratio expression:

\frac{B_{2}}{B_{1}} = \frac{R^{3}}{(R^{2} + (\frac{3}{4} R)^{2})^{3 / 2}} = \frac{R^{3}}{(R^{2} + \frac{9}{16} R^{2})^{3 / 2}} = \frac{R^{3}}{(\frac{16 R^{2} + 9 R^{2}}{16})^{3 / 2}}

Simplify the denominator:

\frac{B_{2}}{B_{1}} = \frac{R^{3}}{(\frac{25 R^{2}}{16})^{3 / 2}} = \frac{R^{3}}{(\frac{25}{16})^{3 / 2} (R^{2})^{3 / 2}} = \frac{R^{3}}{(\frac{5}{4})^{3} R^{3}} = \frac{1}{\frac{125}{64}} = \frac{64}{125}

Thus, the ratio $\frac{B_{2}}{B_{1}}$ is $64 : 125$ .

💡 Teacher's Secret Hint

Pay close attention to the exponent $3 / 2$ and simplify the terms inside the parenthesis first.

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Physics Question 38 – JEE-MAIN 2025

Clue 1 of 3: Recall Magnetic Field Formulas

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