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Numerical · Q15

Q.Taking μ0=4π×10−7 T m A−1\mu_0=4\pi\times10^{-7}\ \text{T m A}^{-1} and ϵ0=8.85×10−12 C2N−1m−2\epsilon_0=8.85\times10^{-12}\ \text{C}^2\text{N}^{-1}\text{m}^{-2}, calculate the speed of an electromagnetic wave in vacuum using c=1/μ0ϵ0c=1/\sqrt{\mu_0\epsilon_0}, and compare your answer with the known speed of light.

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✓ Free question

Given: μ0=4π×10−7 T m A−1=1.2566×10−6 T m A−1\mu_0=4\pi\times10^{-7}\ \text{T m A}^{-1}=1.2566\times10^{-6}\ \text{T m A}^{-1}, ϵ0=8.85×10−12 C2N−1m−2\epsilon_0=8.85\times10^{-12}\ \text{C}^2\text{N}^{-1}\text{m}^{-2}.

Product μ0ϵ0\mu_0\epsilon_0.

μ0ϵ0=(1.2566×10−6)(8.85×10−12)=1.112×10−17\mu_0\epsilon_0 = (1.2566\times10^{-6})(8.85\times10^{-12}) = 1.112\times10^{-17}

Square root.

μ0ϵ0=1.112×10−17≈3.335×10−9\sqrt{\mu_0\epsilon_0} = \sqrt{1.112\times10^{-17}} \approx 3.335\times10^{-9}

Speed.

c=1μ0ϵ0=13.335×10−9≈3.0×108 m/sc = \frac{1}{\sqrt{\mu_0\epsilon_0}} = \frac{1}{3.335\times10^{-9}} \approx 3.0\times10^8\ \text{m/s}

Comparison. The independently measured speed of light in vacuum is 2.998×108 m/s2.998\times10^8\ \text{m/s}, essentially identical (to the precision used here) to the value calculated purely from the electric and magnetic constants μ0\mu_0 and ϵ0\epsilon_0 -- exactly the agreement that led Maxwell to conclude light itself is an electromagnetic wave.

✓Final answer

c=1/μ0ϵ0≈3.0×108 m/sc=1/\sqrt{\mu_0\epsilon_0}\approx3.0\times10^8\ \text{m/s}, matching the measured speed of light in vacuum.

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