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Magnetics question

2006 · Shift 0 · Q91
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Magnetics question

2006 · Shift 0 · Q91

JEE MainPhysicsMagneticsMCQ+4 / −1
A long solenoid has 200200200 turns per cmcmcm and carries a current i.i.i. The magnetic field at its center is 6.28×10−2   Weber/m2.6.28 \times {10^{ - 2}}\,\,\,Weber/{m^2}.6.28×10−2Weber/m2. Another long solenoid has 100100100 turns per cmcmcm and it carries a current i3{i \over 3}3i​. The value of the magnetic field at its center is
  1. A
    1.05×10−2  Weber/m21.05 \times {10^{ - 2}}\,\,Weber/{m^2}1.05×10−2Weber/m2
  2. B
    1.05×10−5  Weber/m21.05 \times {10^{ - 5}}\,\,Weber/{m^2}1.05×10−5Weber/m2
  3. C
    1.05×10−3  Weber/m21.05 \times {10^{ - 3}}\,\,Weber/{m^2}1.05×10−3Weber/m2
  4. D
    1.05×10−4  Weber/m21.05 \times {10^{ - 4}}\,\,Weber/{m^2}1.05×10−4Weber/m2
View written solutionFree

Correct answer: A

  1. Magnetic field inside a long solenoid

    For a long solenoid, B=μ0nIB = \mu_0 n IB=μ0​nI where:

    • μ0\mu_0μ0​ is permeability of free space,
    • nnn is number of turns per unit length,
    • III is current.

  2. For the first solenoid

    Given:

    • n1=200n_1 = 200n1​=200 turns/cm
    • current =i= i=i
    • magnetic field B1=6.28×10−2 Wb/m2B_1 = 6.28 \times 10^{-2}\,\text{Wb/m}^2B1​=6.28×10−2Wb/m2

    So, B1=μ0n1iB_1 = \mu_0 n_1 iB1​=μ0​n1​i

  3. For the second solenoid

    Given:

    • n2=100n_2 = 100n2​=100 turns/cm
    • current =i3= \dfrac{i}{3}=3i​

    Therefore, B2=μ0n2(i3)B_2 = \mu_0 n_2 \left(\frac{i}{3}\right)B2​=μ0​n2​(3i​)

  4. Take ratio of the two fields

    B2B1=μ0n2(i/3)μ0n1i=n23n1\frac{B_2}{B_1} = \frac{\mu_0 n_2 (i/3)}{\mu_0 n_1 i} = \frac{n_2}{3n_1}B1​B2​​=μ0​n1​iμ0​n2​(i/3)​=3n1​n2​​

    Substitute values: B2B1=1003×200=16\frac{B_2}{B_1} = \frac{100}{3 \times 200} = \frac{1}{6}B1​B2​​=3×200100​=61​

    Hence, B2=B16=6.28×10−26B_2 = \frac{B_1}{6} = \frac{6.28 \times 10^{-2}}{6}B2​=6B1​​=66.28×10−2​

  5. Calculate

    B2=1.0467×10−2 Wb/m2B_2 = 1.0467 \times 10^{-2}\,\text{Wb/m}^2B2​=1.0467×10−2Wb/m2

    Approximately, B2≈1.05×10−2 Wb/m2B_2 \approx 1.05 \times 10^{-2}\,\text{Wb/m}^2B2​≈1.05×10−2Wb/m2

  6. Match with options

    This corresponds to: A\boxed{\text{A}}A​

Final Answer: 1.05×10−2 Wb/m21.05 \times 10^{-2}\,\text{Wb/m}^21.05×10−2Wb/m2

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