Find Electric Field Vector for Time-Dependent EM Plane Wave

[lam58]

Q) A harmonic time-dependent electromagnetic plane wave, of angular frequency ω, propagates along the positive z-direction in a source-free medium with σ = 0, ε = 1 and µ = 3. The magnetic field vector for this wave is: H = H_y u_y. Use Maxwell’s equations to determine the corresponding electric field vector.Ans) I've pretty much forgotten all this stuff from 1st year, so I'm not sure if my answer is correct.

$$\bigtriangledown \times H = \varepsilon_0 \frac{\partial E}{\partial t}​$$, $$H = (0, Hy, 0)$$

$$\bigtriangledown \times H = \begin{vmatrix} \hat{u}x & \hat{u}y & \hat{u}z \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 0 & Hy & 0 \end{vmatrix}$$

$$= \hat{u}x (\frac{-\partial Hy}{\partial z}) - \hat{u}y(0) + \hat{u}z (\frac{\partial Hy}{\partial z}) = (\frac{-\partial Hy}{\partial z}, 0, 0)$$

and

$$\varepsilon_0 \frac{\partial E}{\partial t}​ = (\varepsilon_0 \frac{\partial E}{\partial t}​, 0, 0)$$

$$\Rightarrow \varepsilon_0 \frac{\partial E}{\partial t} = \frac{-\partial Hy}{\partial z}$$

At this point I'm somewhat lost as how to find E vector. I know that J = σ E and σ = 0, but how do I get from what I got above to the E vector?

 

[TSny]

Looks OK so far. Maybe bringing in another one of Maxwell's equations will help.

 

[lam58]

Looks OK so far. Maybe bringing in another one of Maxwell's equations will help.

Any suggestions?

 

[rude man]

Heard of "intrinsic impedance"?

If you have to derive this I would start with ∇xE = -∂B/∂t.

 

[TSny]

Out of the 3 other equations, which one do you think would be helpful in relating E and H?

 

[lam58]

Out of the 3 other equations, which one do you think would be helpful in relating E and H?

Heard of "intrinsic impedance"?

If you have to derive this I would start with ∇xE = -∂B/∂t.

Hi, thanks for your help.

Going by what you say, I think then because $$\frac{{\partial {E_x}}}{{\partial z}}{u_y} = - \frac{{\partial B}}{{\partial t}} = -\mu \mu_0 \frac{ \partial Hy}{\partial t}$$,

if I take the partial derivatives of both (one with respect to t and the other to z), I get $$\varepsilon_0 \frac{\partial^2 E_x}{\partial t^2} = \frac{\partial^2 H_y}{\partial t \partial z}$$

and

$$\mu \mu_0 \frac{\partial^2 H_y}{\partial t \partial z} = -\frac{\partial^2 E_x}{\partial z^2}$$.

This then give $$\mu \mu_0 \varepsilon_0 \frac{\partial^2 E_x}{\partial t^2} = \frac{\partial^2 E_x}{\partial z^2} \Rightarrow \frac{\partial^2 E_x}{\partial t^2} = \frac{1}{\mu \mu_0 \varepsilon_0} \frac{\partial^2 E_x}{\partial z^2}$$

Subbing in the numbers for $$\mu \mu_0 \varepsilon_0$$

$$\frac{\partial^2 E_x}{\partial t^2} = 3\times 10^{16} \frac{\partial^2 E_x}{\partial z^2}$$

 

[TSny]

OK. (It might be better to hold off on plugging in numbers.) Use this result with the fact that the time dependence is harmonic with angular frequency ω to deduce how E depends on z.

 

[lam58]

OK. (It might be better to hold off on plugging in numbers.) Use this result with the fact that the time dependence is harmonic with angular frequency ω to deduce how E depends on z.

Ok: $$\frac{\partial^2 E_x}{\partial t^2} = \mu \mu_0 \varepsilon_0 (- k^2 \varepsilon E_x)$$

$$\Rightarrow E_x(z) = \mu \mu_0 \varepsilon_0 A_0 exp[-jkz]$$

$$= 3\times 10^{16} A_0 exp[-jkz]$$

Where $$k = \frac{\omega}{c} = \omega [\mu_0 \varepsilon_0]^{\frac{1}{2}}$$

 

[TSny]

I'm not quite following your logic. (Maybe you just didn't write out the steps.) You are given that E is harmonic in time with angular frequency ω: E_x ~ exp(jωt).

So, ∂²E_x/∂t² = -ω²E_x.

Using this, you can get a differential equation for E_x as a function of z.

 

[lam58]

I'm not quite following your logic. (Maybe you just didn't write out the steps.) You are given that E is harmonic in time with angular frequency ω: E_x ~ exp(jωt).

So, ∂²E_x/∂t² = -ω²E_x.

Using this, you can get a differential equation for E_x as a function of z.

But isn't that similar to what I already had where $$\frac{\partial^2 E_x}{\partial z^2} = -j\omega^2 Ex$$

$$\Rightarrow \frac{\partial^2 E_x}{\partial t^2} = \frac{1}{\mu \mu_0 \varepsilon_0} A_0 exp[-j \omega z]$$

 

[TSny]

But isn't that similar to what I already had where $$\frac{\partial^2 E_x}{\partial z^2} = -j\omega^2 Ex$$

This equation isn't correct.

Go back to what you got from Maxwell's equations:$$\frac{\partial^2 E_x}{\partial t^2} = \frac{1}{\mu \mu_0 \varepsilon_0} \frac{\partial^2 E_x}{\partial z^2}$$
From the fact that $E_x$ is given to vary harmonically in time with angular frequency $\omega$, you can replace the left side of the above equation by $-\omega^2 E_x$. This will give you the differential equation to solve for finding the z-dependence of $E_x$.

 

[lam58]

This equation isn't correct.

Go back to what you got from Maxwell's equations:$$\frac{\partial^2 E_x}{\partial t^2} = \frac{1}{\mu \mu_0 \varepsilon_0} \frac{\partial^2 E_x}{\partial z^2}$$
From the fact that $E_x$ is given to vary harmonically in time with angular frequency $\omega$, you can replace the left side of the above equation by $-\omega^2 E_x$. This will give you the differential equation to solve for finding the z-dependence of $E_x$.

I know the answer is probably really obvious but I'm lost now.

EDIT: I want to say this has something to do with helmholtz equation?

 

[TSny]

You are looking for the electric field $\vec{E}(z, t)$ which you already know only has an x component. So, you are looking for $E_x(z, t)$.

You are given the t dependence as harmonic with frequency $\omega$. So, you can write $E_x(z, t) = E_x(z)e^{-j\omega t}$.

Thus, you must now determine the function $E_x(z)$.

You have $$\frac{\partial^2 E_x}{\partial t^2} = \frac{1}{\mu \mu_0 \varepsilon_0} \frac{\partial^2 E_x}{\partial z^2}$$
What do you get when you substitute $E_x(z, t) = E_x(z)e^{-j\omega t}$ into both sides of this equation and simplify?

 

[rude man]

OP, you're in excellent hands with TSny, but if you want an alternative you could use the equation in post 4 and come up quickly with a 1st-order diff. eq., easily solved if you remember that your B field goes as B₀cos(kz - wt) where k = ω/v and v is the phase velocity of the wave in the z direction.

The catch here is that you need to know v as a function of the basic parameters ε and μ. Which you could easily look up but then you'd lose the insight you gain by pursuing TSny's approach.

 

[TSny]

I might have misinterpreted the problem. I was assuming that the only thing you know is that the time dependence of the wave is harmonic, but that you know nothing of the spatial dependence. Then you can use Maxwell's equations to deduce the spatial dependence and the value of the wavevector k.

But, if "harmonic wave" is interpreted as already having the form cos(kz-ωt), then you just need to determine k. No need to solve a differential equation.

Sorry if I have misinterpreted what you are meant to do. Thanks rude man.

 

[lam58]

OP, you're in excellent hands with TSny, but if you want an alternative you could use the equation in post 4 and come up quickly with a 1st-order diff. eq., easily solved if you remember that your B field goes as B₀cos(kz - wt) where k = ω/v and v is the phase velocity of the wave in the z direction.

The catch here is that you need to know v as a function of the basic parameters ε and μ. Which you could easily look up but then you'd lose the insight you gain by pursuing TSny's approach.

I might have misinterpreted the problem. I was assuming that the only thing you know is that the time dependence of the wave is harmonic, but that you know nothing of the spatial dependence. Then you can use Maxwell's equations to deduce the spatial dependence and the value of the wavevector k.

But, if "harmonic wave" is interpreted as already having the form cos(kz-ωt), then you just need to determine k. No need to solve a differential equation.

Sorry if I have misinterpreted what you are meant to do. Thanks rude man.

I appreciate both of your help, but I reckon I really need to study this more. I'm going to try and catch my lecturer tomorrow and ask. Fortunately this isn't homework, the unfortunate thing is this may potentially be a question in an exam I have in a few days.

I'm an elec eng student and this is from a radio transmissions module tutorial compromising of both the engineering aspects and the physics. Unfortunately I spent too much time studying the engineering part and not enough on the physics.

The thing is I know that I can do this, it feels like I have it in my head I just can't get at it.

Anyway thanks. :)

 

[rude man]

I might have misinterpreted the problem. I was assuming that the only thing you know is that the time dependence of the wave is harmonic, but that you know nothing of the spatial dependence. Then you can use Maxwell's equations to deduce the spatial dependence and the value of the wavevector k.

But, if "harmonic wave" is interpreted as already having the form cos(kz-ωt), then you just need to determine k. No need to solve a differential equation.

Sorry if I have misinterpreted what you are meant to do. Thanks rude man.

Hi T --- if he/she has k you still need to express k (or v = ω/k) in terms of ε and μ the way you indicated, or something similar, right? ε ad μ are introduced via ∇²E = με∂²E/∂t² far as I know. Seems to me you need to wind up with ε and (especially!) μ to answer the question.

 

[TSny]

Hi T --- if he/she has k you still need to express k (or v = ω/k) in terms of ε and μ the way you indicated, or something similar, right? ε ad μ are introduced via ∇²E = με∂²E/∂t² far as I know. Seems to me you need to wind up with ε and (especially!) μ to answer the question.

Yes, that sounds right. You should be able to show k = (ω/c)√μ.