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Magnetics question

2005 · Shift 0 · Q125
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Magnetics question

2005 · Shift 0 · Q125

JEE MainPhysicsMagneticsMCQ+4 / −1
Two concentric coils each of radius equal to 2πcm2\pi cm2πcm are placed at right angles to each other. 333 ampere and 444 ampere are the currents flowing in each coil respectively . The magnetic induction in Weber /m2{m^2}m2 at the center of the coils will be (μ=4π×10−7Wb/A.m)\left( {\mu = 4\pi \times {{10}^{ - 7}}Wb/A.m} \right)(μ=4π×10−7Wb/A.m)
  1. A
    10−5{10^{ - 5}}10−5
  2. B
    12×10−512 \times {10^{ - 5}}12×10−5
  3. C
    7×10−57 \times {10^{ - 5}}7×10−5
  4. D
    5×10−55 \times {10^{ - 5}}5×10−5
View written solutionFree

Correct answer: D

  1. Magnetic field at the center of a circular coil

For a single circular coil of radius RRR carrying current III, the magnetic field at the center is

B=μ0I2RB=\frac{\mu_0 I}{2R}B=2Rμ0​I​

Given:

  • R=2π cm=2π×10−2 mR=2\pi\ \text{cm}=2\pi\times 10^{-2}\ \text{m}R=2π cm=2π×10−2 m
  • μ0=4π×10−7 Wb/A⋅m\mu_0=4\pi\times 10^{-7}\ \text{Wb/A·m}μ0​=4π×10−7 Wb/A⋅m

So,

B=4π×10−7I2⋅2π×10−2B=\frac{4\pi\times 10^{-7} I}{2\cdot 2\pi\times 10^{-2}}B=2⋅2π×10−24π×10−7I​

B=4πimes10−7I4πimes10−2=I×10−5 TB=\frac{4\pi imes 10^{-7} I}{4\pi imes 10^{-2}}=I\times 10^{-5}\ \text{T}B=4πimes10−24πimes10−7I​=I×10−5 T

Thus for each coil:

  • For I1=3I_1=3I1​=3 A, B1=3×10−5 TB_1=3\times 10^{-5}\ \text{T}B1​=3×10−5 T
  • For I2=4I_2=4I2​=4 A, B2=4×10−5 TB_2=4\times 10^{-5}\ \text{T}B2​=4×10−5 T
  1. Direction of the two magnetic fields

The coils are at right angles to each other, so the planes of the coils are perpendicular. Therefore, the magnetic fields at the common center are also perpendicular.

Hence resultant field is

B=B12+B22B=\sqrt{B_1^2+B_2^2}B=B12​+B22​​

B=(3×10−5)2+(4×10−5)2B=\sqrt{(3\times 10^{-5})^2+(4\times 10^{-5})^2}B=(3×10−5)2+(4×10−5)2​

B=10−59+16B=10^{-5}\sqrt{9+16}B=10−59+16​

B=10−5×5=5×10−5 TB=10^{-5}\times 5=5\times 10^{-5}\ \text{T}B=10−5×5=5×10−5 T

  1. Matching with options

5×10−5 Wb/m25\times 10^{-5}\ \text{Wb/m}^25×10−5 Wb/m2

So the correct option is D.

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