Magnetic field energy and force of an electromagnet

[bitrex]

I'm reading up on the equations for electromagnets, and in looking over the Wikipedia page on the subject I'm wondering if it's in error or if there is a flaw in my understanding. Specifically, it says that the force exerted by a magnetic field in an electromagnet on a section of core material is $$\frac{B^{2}A}{2\mu_0}$$. The page then goes on to use this equation to find the force an electromagnet has on say a piece of iron that it is lifting, in a closed magnetic circuit with no air gap. Wouldn't using that equation for a closed magnetic circuit in iron be incorrect? Since (as I understand it) the force on a section of the core is given by the gradient of the magnetic field energy, which is dependent on the permeability of the material in which the magnetic field resides.

I have seen the above equation used elsewhere when the force on the armature of a relay separated from the core by an air gap is calculated. In that case the magnetic field energy is assumed to be zero outside the gap because the core material is high permittivity, and the force is just the derivative of the field energy with respect to the length of the gap. When the iron the electromagnet is lifting actually comes in contact with the electromagnet and completes the magnetic circuit, wouldn't the force exerted reduce as well, since there is less magnetic energy available in a magnetic circuit with no air gap? As you can see, my understanding is a bit shakey and I'd appreciate any clarification.

 

[eljose79]

Hello,

Thank you for bringing up this question about the equations for electromagnets. I can understand why you may be confused about the use of the equation \frac{B^{2}A}{2\mu_0} in a closed magnetic circuit with iron.

First, it is important to note that this equation is derived from the Lorentz force law, which states that the force on a charged particle in a magnetic field is equal to the product of its charge, velocity, and the strength of the magnetic field. In the case of an electromagnet, the force on a section of the core material is the sum of the forces on all the individual charged particles within that section.

Now, let's consider the closed magnetic circuit with iron. The presence of the iron in the circuit does not change the strength of the magnetic field, but it does increase the number of charged particles within the circuit. This means that the force on each individual particle may be smaller, but the total force on the section of the core material will still be equal to \frac{B^{2}A}{2\mu_0}. So, the equation is still valid in this scenario.

However, you are correct in pointing out that the force may decrease if the iron comes in contact with the electromagnet and completes the magnetic circuit. This is because the permeability of iron is higher than air, which means that the magnetic field will be more concentrated in the iron, leading to a decrease in the overall force exerted. In this case, the force can be calculated using the derivative of the magnetic field energy with respect to the length of the gap, as you mentioned.

I hope this helps clarify your understanding of the equations for electromagnets. If you have any further questions, please let me know. Keep up the good work in your research!

 

[astrokreng]

Thank you for bringing this to my attention. I can understand your confusion and I am happy to provide some clarification on this topic.

Firstly, let's define what we mean by magnetic field energy. Magnetic field energy is the energy stored in a magnetic field, and it is directly proportional to the square of the magnetic field strength (B) and the volume of the magnetic field (V). This can be expressed mathematically as E = \frac{1}{2}\mu_0 B^2 V, where \mu_0 is the permeability of free space.

Now, the force exerted by a magnetic field on a section of core material is indeed given by the gradient of the magnetic field energy, as you mentioned. This means that the force is proportional to the change in magnetic field energy over a certain distance. However, in the case of a closed magnetic circuit with no air gap, the magnetic field energy is constant throughout the entire circuit. This is because the magnetic field lines are confined within the core material and there is no change in the magnetic field strength or volume. Therefore, the force on the core material remains constant and can be calculated using the equation \frac{B^2 A}{2\mu_0}.

In the case of an electromagnet lifting a piece of iron, the force exerted on the iron is also dependent on the permeability of the iron. This is because the iron has a higher permeability compared to air, which means it can store more magnetic field energy. When the iron comes into contact with the electromagnet, it completes the magnetic circuit and allows for more magnetic field energy to be stored. This means that the force exerted on the iron will increase, as there is more energy available in the magnetic circuit.

I hope this helps to clarify your understanding. It is important to keep in mind that the equations for electromagnets are simplified models and may not account for all real-world factors. It is always best to consult multiple sources and conduct experiments to verify any calculations.