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Derivation Wave Equation for E-field and B-field

Electromagnetic wave (EM wave)
Level 4 (for physics pros)
Level 4 requires the knowledge of vector calculus, (multidimensional) differential and integral calculus. Suitable for advanced students.
Updated by Alexander Fufaev on
Table of contents
  1. Wave equation for the electric field Here we derive the wave equation for the E-field from Maxwell's third equation in vacuum.
  2. Wave equation for the magnetic field Here we derive the wave equation for the B-field from Maxwell's fourth equation in vacuum.

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Electromagnetic wave (EM wave)
Electromagnetic wave with E-field and B-field component.

The goal is to derive the wave equation for the electric field \(\boldsymbol{E}\) and the magnetic field \(\boldsymbol{B}\) using Maxwell's equations in free space.

We use the four Maxwell equations of electrodynamics in charge-free (\(\rho = 0\)) and current-free (\(\boldsymbol{j} = 0\)) space:

Maxwell equations without charges and currents
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Another ingredient necessary for the derivation of the two wave equations is the following relation for the curl of the curl of the vector field \(\boldsymbol{F}\) (double cross product):

Double cross product of a vector field
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The first summand here is the gradient of the divergence of \(\boldsymbol{F}\) and the second summand is the divergence of the gradient of \(\boldsymbol{F}\). This is just a mathematical relation that can be derived.

Wave equation for the electric field

Maxwell's equations 1 in vacuum are coupled differential equations. To get the wave equation for the E-field, we have to decouple the third Maxwell equation. Apply the curl "\(\nabla \times \)" on both sides of the third Maxwell equation:

Double cross product applied to third Maxwell equation
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The time derivative together with the minus sign may be placed before the nabla operator, because the nabla operator ( spatial derivative) does not depend on the time \(t\):

Third Maxwell equation with time derivative shifted to front
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Now we can replace the curl \( \nabla \times \boldsymbol{B} \) of the magnetic field with the help of the fourth Maxwell equation:

Third and fourth Maxwell equations combined
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The time derivative outside the parenthesis can be placed after the magnetic and electric field constants. Two time derivatives can be compactly combined to the second time derivative:

Decoupled third Maxwell equation
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One side of the wave equation is derived, namely the second time derivative of the electric field. Now all that remains is to rewrite the left-hand side into the correct form, as in a wave equation. Use the double cross product 2:

Decoupled third Maxwell equation with eliminated double cross product
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On the left hand side of Eq. 7 contains the divergence \(\nabla \cdot \boldsymbol{E}\) of the electric field. According to the first Maxwell equation, the divergence of the electric field in charge-free space is always zero. Thus Eq. 7 simplifies to:

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Wave equation for the magnetic field

To derive the wave equation for the magnetic field \(\boldsymbol{B}\), we need to decouple Maxwell's fourth equation in Eq. 1. This is done analogously to the E-field. Apply the curl "\(\nabla \times\)" on both sides of the fourth Maxwell equation:

Double cross product applied to fourth Maxwell equation
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Now place the time derivative and the two constants on the right side in front of the Nabla operator:

Fourth Maxwell equation with pulled out time derivative
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Use Maxwell's third equation to substitute the curl \( \nabla \times \boldsymbol{E} \) of the electric field on the right-hand side:

Decoupled fourth Maxwell equation
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The time derivative is combined and the double cross product on the left side is replaced using Eq. 2:

Decoupled fourth Maxwell equation with eliminated double cross product
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The divergence \(\nabla \cdot \boldsymbol{B}\) is zero according to the second Maxwell equation. Thus you get:

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