# Derivation Wave Equation for E-field and B-field

Level 4 (up to Physics M.Sc.)
Level 4 requires the knowledge of vector calculus, (multidimensional) differential and integral calculus. Suitable for advanced students.
Updated by Alexander Fufaev on

## Video - Electromagnetic Wave Equation Simply Explained

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:

Wave equation for the E-field
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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:

Wave equation for the B-field
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