Magnetic field outside a solenoid in 3D in terms of position

[aravindk87]

Magnetic field outside a solenoid in 3D(off axis)

Hey all,
I am modeling an inductive proximity sensor and require the formula for the magnetic field just outside a solenoid off axis. In terms of a particle's position.i.e.(x,y,z). I have searched a lot of places. But most of the results show it to be zero. Please help.

 

[pam]

The magnetic scalar potential, phi_m, for a single loop is given as a Legendre polynomial expansion in "Classical Electromagnetism" by Franklin.
This can be integrated to give the scalar potential for a solenoid.
Then B=-grad phi_m.
That is the simplest way, but it is quite complicated.

 

[astrokreng]

Hello,

Thank you for your question. The magnetic field outside a solenoid in 3D can be calculated using the Biot-Savart law. The formula for the magnetic field at any point (x,y,z) outside a solenoid with a current I and radius R is given by:

B = μ0I(R^2 + z^2) / 2(R^2 + z^2)^(3/2)

Where μ0 is the permeability of free space (4π x 10^-7 N/A^2) and z is the distance from the center of the solenoid.

Off-axis, the formula becomes more complex as it also takes into account the angle of the point from the axis of the solenoid. The full formula is:

B = μ0I(R^2 + z^2) / 2(R^2 + z^2 + 2Rzsinθ)^(3/2)

Where θ is the angle between the point and the axis of the solenoid.

As you can see, the magnetic field off-axis is not zero. It is dependent on the distance from the center of the solenoid, the angle from the axis, and the current in the solenoid. I hope this helps with your modeling. Best of luck!