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In 1864, James Clerk Maxwell published a theory that included an equation for the speed of electromagnetic waves in a vacuum - AQA - A-Level Physics - Question 2 - 2021 - Paper 7

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In 1864, James Clerk Maxwell published a theory that included an equation for the speed of electromagnetic waves in a vacuum. Show that Maxwell's theory agrees with... show full transcript

Worked Solution & Example Answer:In 1864, James Clerk Maxwell published a theory that included an equation for the speed of electromagnetic waves in a vacuum - AQA - A-Level Physics - Question 2 - 2021 - Paper 7

Step 1

Show that Maxwell's theory agrees with the accepted value for the speed of light in a vacuum.

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Answer

Maxwell's equation for the speed of electromagnetic waves is given by:

C=1μ0ϵ0C = \frac{1}{\sqrt{\mu_0 \epsilon_0}}

where:
CC is the speed of light,
μ0\mu_0 is the permeability of free space, and
ϵ0\epsilon_0 is the permittivity of free space.

From the data booklet, we know that:

  • μ0=4π×107H/m\mu_0 = 4\pi \times 10^{-7} \, \text{H/m}
  • ϵ0=8.85×1012F/m\epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m}

Substituting these values into the formula, we calculate:

C=1(4π×107)(8.85×1012)C = \frac{1}{\sqrt{(4\pi \times 10^{-7}) (8.85 \times 10^{-12})}}

Calculating this gives us approximately 2.998×108m/s2.998 \times 10^8 \, \text{m/s}, which aligns with the accepted value of the speed of light in a vacuum.

Step 2

Explain this experiment with reference to Maxwell's model of electromagnetic waves.

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Answer

In this experiment, the radio wave transmitter T generates an oscillating current, which creates an alternating electric field (E-field) and magnetic field (B-field) perpendicular to each other and to the direction of wave propagation, consistent with Maxwell's model of electromagnetic waves.

The oscillating current in T induces a horizontal electric field that propagates through space. As this E-field varies, it induces a magnetic field around the conductive loop aerial D. The varying magnetic field generates an electromotive force (emf) in the loop aerial.

This implies that electromagnetic waves can transmit energy through space, as the oscillating E-field creates a corresponding B-field, enabling energy transfer between the transmitter and the detector via the space between them.

Step 3

Deduce whether this arrangement can be used to measure speed of electromagnetic waves suggested by Maxwell's equation.

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Answer

In the given experimental setup, stationary waves form between the fixed transmitter and the fixed reflector. The distance between them is approximately 12 m and the transmitter frequency is 75 MHz.

The wavelength (λ\lambda) can be calculated using:
λ=Cf\lambda = \frac{C}{f}
where C3×108m/sC \approx 3 \times 10^8 \text{m/s} and f=75×106Hzf = 75 \times 10^6 \text{Hz}.
Calculating gives us:

λ=3×10875×106=4m\lambda = \frac{3 \times 10^8}{75 \times 10^6} = 4 \text{m}

Since the distance between the transmitter and the reflector is greater than half of the wavelength (i.e., greater than 2m), this arrangement can indeed be used to measure the speed of electromagnetic waves as suggested by Maxwell's equation.

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