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Magnetics question

2023 · 30 Jan · Shift 1 · Q52

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Magnetics
2023 · 30 Jan · Shift 1 · Q52

Magnetics question

2023 · 30 Jan · Shift 1 · Q52

JEE MainPhysicsMagneticsMCQ+4 / −1

The magnetic moments associated with two closely wound circular coils

\mathrm{A}

and

\mathrm{B}

of radius

\mathrm{r}_{\mathrm{A}}=10\mathrm{cm}

and

\mathrm{r}_{\mathrm{B}}=20 \mathrm{~cm}

respectively are equal if : (Where

\mathrm{N}_{\mathrm{A}}, \mathrm{I}_{\mathrm{A}}

and

\mathrm{N}_{\mathrm{B}}, \mathrm{I}_{\mathrm{B}}

are number of turn and current of

\mathrm{A}

and

\mathrm{B}

respectively)

A
$4 \mathrm{~N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}}=\mathrm{N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}}$
B
$2 \mathrm{~N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}}=\mathrm{N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}}$
C
$\mathrm{N}_{\mathrm{A}}=2 \mathrm{~N}_{\mathrm{B}}$
D
$\mathrm{N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}}=4 \mathrm{~N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}}$

View written solutionFree

Correct answer: D

For a circular coil, the magnetic moment is

$m = NIA$

where $N$ is number of turns, $I$ is current, and $A$ is area of the coil.

Since the coils are circular,

$A = \pi r^2$

So,

$m = NI\pi r^2$

For coil $A$ :

$m_A = N_A I_A \pi r_A^2$

Given $r_A = 10\text{ cm}$ .

For coil $B$ :

$m_B = N_B I_B \pi r_B^2$

Given $r_B = 20\text{ cm}$ .

Since the magnetic moments are equal,

$N_A I_A \pi r_A^2 = N_B I_B \pi r_B^2$

Substitute the radii:

$N_A I_A \pi (10)^2 = N_B I_B \pi (20)^2$

Cancel $\pi$ :

$N_A I_A (100) = N_B I_B (400)$

Divide by $100$ :

$N_A I_A = 4N_B I_B$

Compare with the options:

A: $4N_A I_A = N_B I_B$ ❌
B: $2N_A I_A = N_B I_B$ ❌
C: $N_A = 2N_B$ ❌
D: $N_A I_A = 4N_B I_B$ ✅

Therefore, the correct option is D.

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