# Finding the magnetic field of a wave from the E field

• zweebna
In summary: Hope this helps!In summary, the conversation discusses using one of Maxwell's equations to determine the associated magnetic field in free space for a monochromatic electromagnetic wave produced by a spherical source. The electric field is given by a formula involving A, r, theta, and phi, and the hint suggests using time harmonics to simplify the integration process.
zweebna

## Homework Statement

Many sources of electromagnetic waves (stars and light bulbs, for example) radiate in all
directions. A simple example of the electric field for a monochromatic electromagnetic wave produced by a spherical source is
$$E(r,\theta,\phi,t)=A\frac{\sin \theta}{r} \big(\cos (kr-\omega t)-\frac{1}{kr} \sin (kr-\omega t) \big) \hat{\phi}$$
where A is a constant and ##k=\omega / c##. Use one of Maxwell's Equations to determine the associated magnetic fi eld in free space (i.e. ##\rho = 0## and ##J = 0##). For simplicity of notation, let ##u = kr - \omega t##. Hint: You will not need to do any difficult integrals.

## Homework Equations

$$\nabla \times E = - \frac{\delta B}{\delta t}$$

## The Attempt at a Solution

I mostly just want to make sure that I'm doing this right. So to find the magnetic field, according to Faraday's law I would take the curl of the electric field and that would be equal to the negative time derivative of the B field, so I can then take a negative integral to get the B field. For ##\nabla \times E## I have
$$\nabla \times E = \frac{1}{r \sin \theta} \bigg [\frac{\delta}{\delta \theta} \big(A \frac{\sin^2 \theta}{r} ( \cos u - \frac{1}{kr} \sin u) \big) \bigg] \hat{r} + \frac{1}{r} \bigg [ - \frac{\delta}{\delta r} \big ( A \sin \theta ( \cos u - \frac{1}{kr} \sin u) \big) \bigg] \hat{\theta}$$
$$\nabla \times E = A \frac{2 \cos \theta}{r^2}( \cos u - \frac{1}{kr} \sin u) \hat{r} - A \frac{ \sin \theta}{r}(-k \sin u + \frac{\sin u - kr \cos u}{k r^2})\hat{\theta}$$

Now assuming I've done this right, then I can take the negative integral to get the B field. I'm a bit confused about this integral though. What would the limits be? Or would it just be an indefinite integral?

Also the hint is making me wary, as this integral seems like it would be kind of difficult. Is there an easier way that I'm missing?

zweebna said:

## Homework Statement

Many sources of electromagnetic waves (stars and light bulbs, for example) radiate in all
directions. A simple example of the electric field for a monochromatic electromagnetic wave produced by a spherical source is
$$E(r,\theta,\phi,t)=A\frac{\sin \theta}{r} \big(\cos (kr-\omega t)-\frac{1}{kr} \sin (kr-\omega t) \big) \hat{\phi}$$
where A is a constant and ##k=\omega / c##. Use one of Maxwell's Equations to determine the associated magnetic fi eld in free space (i.e. ##\rho = 0## and ##J = 0##). For simplicity of notation, let ##u = kr - \omega t##. Hint: You will not need to do any difficult integrals.

## Homework Equations

$$\nabla \times E = - \frac{\delta B}{\delta t}$$

## The Attempt at a Solution

I mostly just want to make sure that I'm doing this right. So to find the magnetic field, according to Faraday's law I would take the curl of the electric field and that would be equal to the negative time derivative of the B field, so I can then take a negative integral to get the B field. For ##\nabla \times E## I have
$$\nabla \times E = \frac{1}{r \sin \theta} \bigg [\frac{\delta}{\delta \theta} \big(A \frac{\sin^2 \theta}{r} ( \cos u - \frac{1}{kr} \sin u) \big) \bigg] \hat{r} + \frac{1}{r} \bigg [ - \frac{\delta}{\delta r} \big ( A \sin \theta ( \cos u - \frac{1}{kr} \sin u) \big) \bigg] \hat{\theta}$$
$$\nabla \times E = A \frac{2 \cos \theta}{r^2}( \cos u - \frac{1}{kr} \sin u) \hat{r} - A \frac{ \sin \theta}{r}(-k \sin u + \frac{\sin u - kr \cos u}{k r^2})\hat{\theta}$$

Now assuming I've done this right, then I can take the negative integral to get the B field. I'm a bit confused about this integral though. What would the limits be? Or would it just be an indefinite integral?

Also the hint is making me wary, as this integral seems like it would be kind of difficult. Is there an easier way that I'm missing?
I did not double check your calculation of the curl but for the integration, you may simply do an indefinite integral. It is not hard since you just integrate with respect to time, so you just need to integrate ##cos u## and ##sin u##, which is easy.

I haven't gone through your math, but in EE we usually attack this problem by using time harmonics.
Your maxwell equation will then become:
∇ x E = -jwB

## What is the relationship between the electric field and the magnetic field in a wave?

The electric and magnetic fields in a wave are perpendicular to each other and are in phase. This means that when the electric field is at its maximum, the magnetic field is also at its maximum, and when the electric field is at its minimum, the magnetic field is also at its minimum.

## How can I calculate the magnetic field of a wave from the electric field?

To calculate the magnetic field of a wave from the electric field, you can use the equation B = E/c, where B is the magnetic field, E is the electric field, and c is the speed of light in a vacuum. This equation is known as the electromagnetic wave equation.

## What is the direction of the magnetic field in relation to the direction of the wave?

The direction of the magnetic field is perpendicular to both the direction of the wave and the direction of the electric field. This means that the magnetic field is always orthogonal to the wave and electric field vectors.

## How do the properties of the medium affect the magnetic field of a wave?

The properties of the medium, such as its permeability and permittivity, can affect the strength and direction of the magnetic field in a wave. In different mediums, the speed of the wave and the magnitude of the magnetic field may vary.

## Can the magnetic field of a wave be measured directly?

No, the magnetic field of a wave cannot be measured directly. It can only be inferred from the electric field using the electromagnetic wave equation. However, the magnetic field can be indirectly measured using specialized equipment, such as a magnetometer.

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