Introductory Electromagnetics: induced E-field

[nonequilibrium]

Homework Statement

"You have an ideal solenoid with n windings per meter, with a current I = I' sin(wt) going through it. Calculate the induced electric field inside and outside the coil. Side: compare with dipole antenna."

Homework Equations

Maxwell equations
B = n mu I (magnetic field inside a perfect coil)

The Attempt at a Solution

Shall I assume that we have to calculate the E-field induced by the changing B-field that is in its turn induced from I? Or is there a more primary E-field I'm forgetting...

So if I imagine a coil laying for me from left to right with the B-field inside at a certain moment pointing to the left, I can take a vertical slice. The E-field at a distance r from the center of the solenoid will be pointing perpendicular on the B-field. Due to symmetry, I can argue that in a circle with a radius r (smaller than the radius of the solenoid) the E field is the same in size all over the circle. So with Faraday's Law:

E 2pi r = d(n mu I pi r²)/dt = n mu I' pi r² w cos(wt)

So inside the solenoid: E = n mu I' r w cos(wt) / 2?

For outside the solenoid, I think I can take B = 0, and then with the same symmetry reasoning, I get E ~ 1/r outside the solenoid.

As for the dipole antenna: they're in principle the same? (causing EM-radiation at a distance) That last question is a bit vague...

A difference between the two is that the E-field with a dipole antenna is more "direct", in a way that the charges themselves generate the E-field. Here the E-field is very secondary, only coming from a changing B-field that was induced itself... Is this an important difference?

 

[anathema]

it is important to carefully consider all aspects of a problem before attempting to solve it. In this case, we are given an ideal solenoid with n windings per meter and a sinusoidal current passing through it. The task is to calculate the induced electric field inside and outside the coil and make a comparison with a dipole antenna.

First, let's consider the primary source of the induced electric field. In this case, it is the changing magnetic field induced by the sinusoidal current passing through the solenoid. This is in line with Faraday's Law, which states that a changing magnetic field will induce an electric field. Therefore, we can proceed with our calculations based on this understanding.

Inside the solenoid, we can use the formula B = nμI to calculate the magnetic field. Then, using Faraday's Law, we can calculate the induced electric field as E = -dΦ/dt. Since the magnetic field is changing in a sinusoidal manner, the induced electric field will also be sinusoidal. Therefore, we can write E = E' sin(wt) where E' is the amplitude of the induced electric field.

To calculate the amplitude, we can use the formula E' = nμI'w, where I' is the amplitude of the current and w is the angular frequency. So inside the solenoid, the induced electric field will be E = nμI'w sin(wt).

Outside the solenoid, we can assume that the magnetic field is negligible and therefore, the induced electric field will be solely due to the changing magnetic field inside the solenoid. Using the same formula as before, we can calculate the amplitude of the induced electric field outside the solenoid as E' = nμI'w. Therefore, outside the solenoid, the induced electric field will be E = nμI'w sin(wt).

Now, let's compare this with a dipole antenna. A dipole antenna also produces an electric field due to a changing magnetic field created by the oscillating charges on the antenna. However, there are some differences between the two. One major difference is that the dipole antenna is designed to radiate electromagnetic waves, whereas the solenoid is not. Additionally, the dipole antenna is more efficient in producing a direct electric field, while the solenoid's induced electric field is more of a secondary