# Calculating the Magnetic Field in Free Space

• miew
In summary, the electric field in free space is E=Eo(x^+y^)sin(2pi/lamda)(z+ct), with Eo=2 statvolts/cm the magnetic field, not including any static magnetic field, must be B=1/c ∂E/∂t.

## Homework Statement

If the electric field in free space is E=Eo(x^+y^)sin(2pi/lamda)(z+ct), with Eo=2 statvolts/cm the magnetic field, not including any static magnetic field, must be what?

2. Relevant equation

∇ x B= 1/c ∂E/∂t

## The Attempt at a Solution

First I calculated ∂E/∂t= cE0/c (x^+y^)cos(2pi/lamba)(x+ct)

Since B is perpendicular to E, it just has a z^ direction.

∇xB=∂yBzx^-∂xBzj^.

Then I equal both equations and I got Bz=Boc(yx^-xj^) cos...

Is this right ?

I assume your hats mean unit vectors? So your E field is propagating in the z direction and the Field vector points in $$\hat x + \hat y$$?

You made some mistake when you calculated $$\partial E/\partial t$$. The argument inside the sine should not change and you get some additional factors.
You first guess for B is wrong. Z is not the only direction perpendicular to E.

Yes, they are unit vector. And the x inside the sin it is supposed to be a z, sorry :/

So aren't there two directions in which B can be perpendicular? x+z and y+z ?

There are two directions but in your first post you gave only one, z. And they are not the ones you gave in your last post.
In addition you only calculated $$B_z=..\hat x$$ as proportional to a vector. But Bz should just be a number.

Unless you know from somewhere else that E and B are perpendicular, so far from the equations you don't see it. In this case you would have to start with a general B.

So then,
∂yBz-∂zBy= Eocos(A) (where A is everything after cos)
∂zBx-∂xBz= Eocos(A)
∂xBy=∂yBx=0

Is that right ? if it is, how do I solve it ?

Yes. The trick is to differentiate again and then combine the equations that you only have one B in the equation. Then you can integrate.

Differentiate with respect to what ? :/

x,y,z. You just have to find the right combination.

Okay, I am going to try that !

Thanks :)

## 1. How do you calculate the magnetic field in free space?

To calculate the magnetic field in free space, you can use the formula B = μ0I/2πr, where B is the magnetic field strength, μ0 is the permeability of free space (4π x 10^-7 H/m), I is the current, and r is the distance from the current.

## 2. What is the unit of measurement for magnetic field in free space?

The unit of measurement for magnetic field strength in free space is tesla (T) or gauss (G).

## 3. How does the magnetic field vary with distance from the current?

The magnetic field strength decreases as the distance from the current increases. This is because the magnetic field follows an inverse square law, meaning that the strength is inversely proportional to the square of the distance.

## 4. Can the magnetic field in free space be affected by external factors?

In free space, the magnetic field is not affected by external factors such as materials or other magnetic fields. However, it can be affected by the presence of a current or a changing electric field.

## 5. What is the significance of calculating the magnetic field in free space?

Calculating the magnetic field in free space is important in understanding and predicting the behavior of electromagnetic waves and the interactions between charged particles. It is also crucial in the design and operation of various devices such as motors, generators, and transformers.