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Electromagnetic Waves question

2022 · 29 Jun · Shift 2 · Q67
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  5. /2022 · 29 Jun · Shift 2 · Q67

Electromagnetic Waves question

2022 · 29 Jun · Shift 2 · Q67

JEE MainPhysicsElectromagnetic WavesNumerical+4 / −1
The displacement current of 4.425 μ\muμ A is developed in the space between the plates of parallel plate capacitor when voltage is changing at a rate of 106 Vs −-− 1. The area of each plate of the capacitor is 40 cm2. The distance between each plate of the capacitor is x ×\times× 10 −-− 3 m. The value of x is ‾\underline{\hspace{2cm}}​. (Permittivity of free space, E0 = 8.85 ×\times× 10 −-− 12 C2 N −-− 1 m −-− 2).
Numerical answer
View written solutionFree

Correct answer: 8

  1. Use displacement current relation

For a parallel plate capacitor, the displacement current is

Id=ε0dΦEdtI_d = \varepsilon_0 \frac{d\Phi_E}{dt}Id​=ε0​dtdΦE​​

Since electric field between plates is uniform,

ΦE=EA=VdA\Phi_E = EA = \frac{V}{d}AΦE​=EA=dV​A

So,

Id=ε0A1ddVdtI_d = \varepsilon_0 A \frac{1}{d}\frac{dV}{dt}Id​=ε0​Ad1​dtdV​

Thus,

d=ε0A dVdtIdd = \frac{\varepsilon_0 A \, \dfrac{dV}{dt}}{I_d}d=Id​ε0​AdtdV​​

  1. Substitute the given values
  • Id=4.425 μA=4.425×10−6 AI_d = 4.425\,\mu A = 4.425 \times 10^{-6}\,AId​=4.425μA=4.425×10−6A
  • ε0=8.85×10−12\varepsilon_0 = 8.85 \times 10^{-12}ε0​=8.85×10−12
  • A=40 cm2=40×10−4=4×10−3 m2A = 40\,cm^2 = 40 \times 10^{-4} = 4 \times 10^{-3}\,m^2A=40cm2=40×10−4=4×10−3m2
  • dVdt=106 Vs−1\dfrac{dV}{dt} = 10^6\,V s^{-1}dtdV​=106Vs−1

Therefore,

d=(8.85×10−12)(4×10−3)(106)4.425×10−6d = \frac{(8.85 \times 10^{-12})(4 \times 10^{-3})(10^6)}{4.425 \times 10^{-6}}d=4.425×10−6(8.85×10−12)(4×10−3)(106)​

  1. Simplify numerator

8.85×4=35.48.85 \times 4 = 35.48.85×4=35.4

and

10−12×10−3×106=10−910^{-12} \times 10^{-3} \times 10^6 = 10^{-9}10−12×10−3×106=10−9

So numerator becomes

35.4×10−9=3.54×10−835.4 \times 10^{-9} = 3.54 \times 10^{-8}35.4×10−9=3.54×10−8

Hence,

d=3.54×10−84.425×10−6d = \frac{3.54 \times 10^{-8}}{4.425 \times 10^{-6}}d=4.425×10−63.54×10−8​

  1. Compute the value

3.544.425=0.8\frac{3.54}{4.425} = 0.84.4253.54​=0.8

and

10−8/10−6=10−210^{-8}/10^{-6} = 10^{-2}10−8/10−6=10−2

Thus,

d=0.8×10−2=8×10−3 md = 0.8 \times 10^{-2} = 8 \times 10^{-3}\,md=0.8×10−2=8×10−3m

Given that distance is x×10−3 mx \times 10^{-3}\,mx×10−3m,

x=8x = 8x=8

  1. Comparison with stored answer

Stored correct answer = 888

Our derived answer also is 888.

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