Homework Help: Find Magnetic Field from Electric Field Using Maxwell's Equations

1. Jul 7, 2014

1. The problem statement, all variables and given/known data
An electromagnetic wave has an electric field $$\mathbf{E} = E_0 \cos(kz-ωt) \hat{x}$$. Using Maxwell's equations, find the magnetic field.

2. Relevant equations
$$\mathbf{∇\times E} = \mathbf{\dot{B}}$$

3. The attempt at a solution

So this problem appears extremely simple, but other students have told me my answer is incorrect, and I can't figure out what is wrong with my math. I find the cross product, which results in the following equation for the time derivative of the magnetic field:

$$\mathbf{\dot{B}} = \hat{y}kE_0\sin(kz-ωt)$$

I now integrate both sides with respect to time. This is where my answer diverges from others, so I'll fully write out my steps:

$$\mathbf{B} = kE_0∫_0^t \sin(kz-ωt') dt'\hat{y}$$

I set $$u = kz-ωt'$$, which means $$du = -ωdt'$$

Plugging this in, the new integral is:

$$\mathbf{B} = -\frac{kE_0}{ω}∫_{kz}^{kz-ωt}\sin(u)du$$

The result is then

$$\mathbf{B} = \frac{kE_0}{ω}[\cos(kz-ωt)-\cos(kz)]\hat{y}$$

However, every student I've talked to has told me that the correct answer should be

$$\mathbf{B} = \frac{kE_0}{ω} \cos(kz-ωt)\hat{y}$$

Is there something simple I'm missing? There's nothing else in the problem description I didn't write. The other answer looks more correct but I can't find any reason that mine is incorrect.

Last edited: Jul 7, 2014
2. Jul 7, 2014

TSny

The left side of the second equation above is incorrect. Think again about what you get when you integrate the left side of the first equation from t' = 0 to t' = t.

Last edited: Jul 7, 2014
3. Jul 8, 2014