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Maxwell's Wave Equation (AQA A Level Physics)

Revision Note

Dan Mitchell-Garnett

Last updated

Nature of Electromagnetic Waves

  • James Clerk Maxwell was a Scottish physicist who, in 1864, published a paper relating electric and magnetic fields

    • These included a series of equations which predicted the existence of oscillating electric and magnetic fields which propagated each other, called electromagnetic waves

  • A charged particle has an electric field

    • An accelerating charge produces an electric field which alternates perpendicular to the particle's motion

    • That alternating electric field produces a perpendicular alternating magnetic field

    • The alternating magnetic field produces an alternating electric field and so on - this is called self-propagation and is why light does not need a medium to travel

Diagram showing the alternating magnetic and electric fields

EM wave

In an electromagnetic wave, the electric (E) field's oscillation generates a perpendicular magnetic (B) field which is also oscillating.

Maxwell's Formula for Electromagnetic Waves

  • One consequence of Maxwell's equations was a prediction of the speed of electromagnetic waves in a vacuum, c :

c space equals space fraction numerator 1 over denominator square root of mu subscript 0 epsilon subscript 0 end root end fraction

  • Here, μ0 is the permeability of free space, a constant

    • ε0 is the permittivity of free space, also a constant (which has already been encountered in the Electric Fields topic)

    • Both of these values are constant, so the speed of electromagnetic waves in a vacuum is constant

  • The permeability of free space, μ0 is a constant that relates magnetic flux density with the current in free space that produces the field

  • The permittivity of free space, ε0 similarly is a constant that relates electric field strength with the charged object in free space producing the electric field

Worked Example

Maxwell calculated his value of the speed of electromagnetic waves, c, using values for ε0 and μ0 which were both well known at the time. In 1855, Weber and Kohlrausch measured the speed of light to be 3.107 × 108 m s−1.

Calculate the difference in Maxwell's value and Weber and Kohlrausch's value of the speed of light.

Give your answer as a percentage of Maxwell's value. 

Answer:

Step 1: List the known quantities:

  • Permittivity of free space, ε0 = 8.85 × 10−12 F m−1

  • Permeability of free space, μ0 = 4π × 10−7 H m−1

  • Weber and Kohlrausch's value of the speed of light, cWK = 3.107 × 108 m s−1 

Step 2: Calculate Maxwell's value of c :

  • Substitute the values for permittivity and permeability into Maxwell's equation for c :

c space equals space fraction numerator 1 over denominator square root of mu subscript 0 epsilon subscript 0 end root end fraction space equals space fraction numerator 1 over denominator square root of open parentheses 4 straight pi space cross times space 10 to the power of negative 7 end exponent close parentheses space cross times space open parentheses 8.85 space cross times space 10 to the power of negative 12 end exponent close parentheses end root end fraction

c space equals space 2.999 space cross times space 10 to the power of 8 space straight m space straight s to the power of negative 1 end exponent

Step 3: Calculate the difference between this and Weber and Kohlrausch's value:

  • Subtract Maxwell's value from cWK :

c subscript W K end subscript space minus space c space equals space open parentheses 3.107 space cross times space 10 to the power of 8 space close parentheses space minus space open parentheses 2.999 space cross times space 10 to the power of 8 close parentheses space equals space 1.08 space cross times space 10 to the power of 7 space straight m space straight s to the power of negative 1 end exponent

Step 4: Calculate this as a percentage of c :

  • Divide this difference by c  and multiply by 100% for a percentage:

fraction numerator c subscript W K end subscript space minus space c over denominator c end fraction space cross times space 100 percent sign space equals space fraction numerator 1.08 space cross times space 10 to the power of 7 over denominator 2.999 space cross times space 10 to the power of 8 end fraction space cross times space 100 percent sign space equals space 3.60 space percent sign

  • The difference in the values is 3.60 % of Maxwell's value

Examiner Tips and Tricks

Make sure you understand what the permittivity and permeability of free space relate to - these are easy to mix up with each other. If you are unsure, check your data booklet. The symbols are listed there with their names, so you only need to remember that ε  (epsilon) is electric and μ  (mu) is magnetic.

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    Dan Mitchell-Garnett

    Author: Dan Mitchell-Garnett

    Expertise: Physics Content Creator

    Dan graduated with a First-class Masters degree in Physics at Durham University, specialising in cell membrane biophysics. After being awarded an Institute of Physics Teacher Training Scholarship, Dan taught physics in secondary schools in the North of England before moving to Save My Exams. Here, he carries on his passion for writing challenging physics questions and helping young people learn to love physics.