# How to prove that H_a and H_b are orthogonal?

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1. Dec 20, 2015

### flux!

1. Okay, so I am gonna prove that

$$\int H_a\cdot H_bdv=0$$

Hint: Use vector Identities

H is the Magnetic Field and v is the volume.

2. Relevant equations this this

$$k_bH_b=\nabla \times E_b$$
$$k_aH_a=\nabla \times E_a$$

k is the wave vector and E is the electric field

3. The attempt at a solution
It is known that H_a and H_b are really perpendicular to each other, so their dot product is just simple, zero. Well I am dead wrong! I got only 2 points out of 10, so its definitely not the solution for proving it.

Its now semester break, so I could not ask our professor the solution (no classes now), plus he is too busy. But I am still itching to find the correct solution for this. The hint tells to use vector identities, how could I figure It out?

2. Dec 21, 2015

### blue_leaf77

Please write the complete description of the problem.

3. Dec 21, 2015

### Fredrik

Staff Emeritus
Also, please explain your "relevant equations". I don't know what you mean by $E_a$ and $E_b$. If they are two different components of the electric field, then I can't make sense of the notation $\nabla\times E_a$. If it's too much work to include a derivation of your relevant equations from Maxwell's equations, then you should at least tell us where you found them.

4. Dec 21, 2015

### flux!

Complete Question:

Prove that H_a and H_b are orthonormal by showing

$$\int H_a \cdot H_b = 0$$

Hint: You may use Vector Identities

---End of Complete Question that is all in there---

The equation:

$$k_aH_a = \nabla \times E_a$$

were just another given equation, could be used or not (He always gives us tricky questions)

E_a and E_b were two different electric field that concurrently exist with H_a and H_b respectively.

5. Dec 21, 2015

### blue_leaf77

You can at least say in which context this question was raised. Your last post didn't narrow down the infinite number of possibilities of how the two fields can be arbitrarily generated and oriented.

6. Dec 21, 2015

### flux!

Do you have your own version by employing vector identities?

7. Dec 21, 2015

### Fredrik

Staff Emeritus
I still don't get it. The problem is asking you to prove a statement about two vector fields $H_a$ and $H_b$, but it doesn't define $H_a$ and $H_b$. So what definition of $H_a$ and $H_b$ are you supposed to use?

You said that $k_aH_a=\nabla\times E_a$ is just another equation that you may or may not need. So that's not the definition of $H_a$, right? Because you absolutely have to use the definition of $H_a$.

You said that $k_a$ and $k_b$ are vectors before. Have you changed your mind about that? How are you making sense of the product $k_aH_a$ if both factors are vectors?

8. Dec 21, 2015

### blue_leaf77

In most topics, it is usually meant to be the magnitude of the wavevector of a lightwave. Therefore those k's must be scalars. Anyway, the given information is indeed lacking in the additional relationship between $H_a$ and $H_b$.