Time-varying magnetic field outside a solenoid

[trix312]

something that has had me confused:

How can I calculate the time varying magnetic field produced by a solenoid, outside the solenoid along the axis of the core. I know that:

inside the solenoid : apply amperes law
For DC (magnetostatic) case outside the solenoid : apply Biot-savart law

but what about a AC (time varying field) outside the solenoid? is the attenuation calculated like in EM waves attenuation and skin-depth? I also read something about replacing biot-savart with Jefimenko's equations if magnetostatics does not apply. Or is this not possible to calculate analytically.

What is going on?

 

[eljose79]

Calculating the time varying magnetic field produced by a solenoid outside the solenoid along the axis of the core can be a complex and confusing task. However, there are several approaches that can help you understand and calculate this field.

As you mentioned, for the DC (magnetostatic) case, you can apply Ampere's law inside the solenoid. This law states that the line integral of the magnetic field around a closed loop is equal to the current passing through that loop. This approach works well for steady currents, but for time varying fields, it becomes more complicated.

For the AC case, you can use the Biot-Savart law, which relates the magnetic field at a point to the current element at that point. This law can be used to calculate the magnetic field outside the solenoid along the axis of the core. However, this approach also becomes more complicated for time varying fields, as it requires calculating the magnetic field at each point along the axis.

Another approach is to use Jefimenko's equations, which take into account the time varying nature of the field. These equations can be used for both the inside and outside of the solenoid, and they take into account the displacement current, which is important for time varying fields. However, these equations can be quite complex and may not be easily solvable analytically.

In terms of attenuation, the approach will depend on the frequency of the time varying field. For low frequencies, the attenuation can be calculated using the skin depth, which is the depth at which the amplitude of the field is reduced by a factor of e. For higher frequencies, the attenuation will depend on the material properties and the frequency of the field.

In summary, calculating the time varying magnetic field outside a solenoid can be a challenging task. Depending on the frequency and complexity of the field, different approaches can be used, such as Ampere's law, Biot-Savart law, or Jefimenko's equations. It is important to consider the displacement current and material properties when calculating the attenuation of the field. I hope this helps clarify some of your confusion.