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JEE Physics 2026 Question with Solution

An electromagnetic wave of frequency 100 MHz100 \, \text{MHz} propagates through a medium of conductivity, σ=10 mho/m\sigma = 10 \, \text{mho/m}. The ratio of maximum conduction current density to maximum displacement current density is \hspace{1cm. [Take 14πϵ0=9×109 N m2/C2]\left[ \text{Take } \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \right]

Answer

Correct answer:1800

Step-by-step solution

Standard Method

Given: frequency f=100 MHz=108 Hzf = 100 \, \text{MHz} = 10^8 \, \text{Hz} and conductivity σ=10 mho/m\sigma = 10 \, \text{mho/m}.

Find: the ratio of maximum conduction current density to maximum displacement current density.

For a sinusoidal electromagnetic wave,

Jc,max⁡=σE0J_{c,\max} = \sigma E_0

and

Jd,max⁡=ϵ0ωE0J_{d,\max} = \epsilon_0 \omega E_0

Therefore,

Jc,max⁡Jd,max⁡=σϵ0ω=σϵ0(2πf)\frac{J_{c,\max}}{J_{d,\max}} = \frac{\sigma}{\epsilon_0 \omega} = \frac{\sigma}{\epsilon_0 (2\pi f)}

Using

14πϵ0=9×109\frac{1}{4\pi\epsilon_0} = 9 \times 10^9

we get

ϵ0=136π×109\epsilon_0 = \frac{1}{36\pi \times 10^9}

Also,

ω=2πf=2π×108 rad/s\omega = 2\pi f = 2\pi \times 10^8 \, \text{rad/s}

Now,

ϵ0ω=(136π×109)(2π×108)=1180\epsilon_0 \omega = \left(\frac{1}{36\pi \times 10^9}\right)(2\pi \times 10^8) = \frac{1}{180}

Hence,

Jc,max⁡Jd,max⁡=101/180=1800\frac{J_{c,\max}}{J_{d,\max}} = \frac{10}{1/180} = 1800

Therefore, the ratio of maximum conduction current density to maximum displacement current density is 18001800.

The solution's lists 180180 as the correct answer, but the extracted solution working gives 18001800.

Common mistakes

  • Using ff directly in place of ω\omega is incorrect because displacement current density depends on ω=2πf\omega = 2\pi f. Always convert frequency to angular frequency before substitution.

  • Missing the factor of conductivity σ=10\sigma = 10 at the final step gives 180180 instead of the correct ratio. After finding ϵ0ω=1180\epsilon_0 \omega = \frac{1}{180}, divide σ\sigma by this quantity.

  • Confusing conduction current density with displacement current density is incorrect because Jc,max⁡=σE0J_{c,\max} = \sigma E_0 whereas Jd,max⁡=ϵ0ωE0J_{d,\max} = \epsilon_0 \omega E_0. Write both expressions separately before taking the ratio.

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