The difference between Vector Helmholtz and Scalar Helmholtz equations.

While studying EM wave propagation we derive the wave equations starting from the Maxwell’s equations. In the process we are encountered with Helmholtz equations. Let’s see, the difference between Vector Helmholtz and Scalar Helmholtz equations.

You can watch the quick video below or read along.

EM wave phenomenon could be derived with the help of Maxwell’s equations. For simplicity we can assume medium as the free space.

So, starting from the wave equations we simplify them to reduce into the equation shown in diagram below.

This equation, \nabla^2{\overrightarrow E}_s=-\beta^2{\overrightarrow E}_s is known as Vector Helmholtz Equation.

Now, when we simplfy this by writing the component of E and comparing both the sides of this equation, then we get three different equations involving each component of E.

These 3 equations one for each component of E are known as Scalar Helmholtz Equations.

Vector Helmholtz Equation and Scalar Helmholtz Equation

Remember

• Vector Helmholtz Equation consists of laplacian (\nabla^2) operation acting on the vector function.
• But, you know that laplacian (\nabla^2) operation is performed over the vector function.
• Each Vector Helmholtz Equation gives us 3 Scalar Helmholtz Equations. One for each component.