Why do magnetic fields store energy

The sink

(Course level > Electro-magnetism)

A spool consists of a cable wound in loops. The cable is often wound on a core made of iron or ferrite.

  • A cable wrapped around a plastic tube nine times is already a solenoid.

  • The large, flat-wound coil of an induction hob. There are several toroidal coils with ferrite or iron cores on the circuit boards of the control electronics.

  • In this transformer, two cylinder coils share the same iron core.

  • The coil of this electromagnet is superconducting at low temperatures. If you "push" the current once, it continues to flow unchecked.

  • This solenoid has only 7 turns. The magnetic field is mainly located in the current-carrying coil, which can be clearly seen from the aligned iron filings.

  • This representation of the field lines and field areas shows the approximate homogeneity of the magnetic field within the current-carrying coil.

Three different aspects

Coils are important in theory and practice from various points of view:

The homogeneous magnetic field of a current-carrying coil

In the case of a "slim" cylinder coil without an iron core, one can assume that the magnetic field is only located within the coil. The field is then homogeneous in terms of direction and strength. Therefore, the magnetic field of the coil plays a major role in many theoretical considerations.
For the electric field, consider the field of a capacitor.
In this way one can examine the energy content of the magnetic field and also make statements about the behavior of matter in the magnetic field[1] do.

The coil as a current and energy store

Magnetic fields can store energy. If you pull the north and south poles of two fixed magnets apart, the energy required for this is stored in the magnetic field. The field can pull the two poles together again, releasing the stored energy again. We have also made this qualitative consideration for the electrical and gravitational fields.
In electrical circuits, coils have the task of temporarily storing energy in their magnetic field and possibly in their iron core. Because coils react primarily to changes in the current, they play a major role in AC technology as a kind of "inertia of the current".
Normal coils are unsuitable for storing energy in large quantities, because the ohmic resistance of the cable heats up the coil and the energy "is lost". If special superconducting cables are used, this form of energy "loss" is eliminated, which is why it is used as a superconducting magnetic energy store.

Coils as "receivers" of induction voltage

Particularly in the case of coils with many turns, a changing magnetic flux induces a voltage through the coil, which is used in many technical devices.

Comparison of capacitor and coil

The quantitative descriptions of a "slim" cylinder coil have many parallels to a capacitor and also have significant differences. The comparison makes it easy to see the properties of both components.

similarities
  • A capacitor generates an electric field when a voltage is applied.
  • The coil generates a magnetic field through a current flow.
  • The greater the voltage of the capacitor, the more
    • greater the electric field strength
    • greater the electrical flux ("number of field lines") and the charge shift
  • The greater the current through the coil, the more
    • greater the magnetic field strength
    • greater the magnetic flux ("number of field lines")
  • The capacitor stores energy
    • in the electric field and
    • possibly in the electrically polarized dielectric
  • The coil stores energy
    • in the magnetic field and
    • possibly in the magnetized iron core
  • The capacitor properties are determined by:
    • the plate area
    • the plate spacing
    • the dielectric
  • The coil properties are determined by:
    • the area of ​​the turns
    • the length of the coil
    • the coil core
differences
  • The electric field in the capacitor is a source field.
    • Field lines begin with positive charges and end with negative ones.
    • A potential (in volts) can be assigned to the field areas.
  • The coil's magnetic field is a vortex field.
    • The field lines are closed and enclose the electric current.
    • The field areas cannot be assigned unambiguous values ​​for a potential.
  • In the case of a coil, the current density or the sensitivity to induction voltages can be increased by increasing the number of turns.
Formulas
capacitor Kitchen sink

creates an electric field,
the strength of which depends on the tension and the distance between the plates.

[math] E = \ frac {U} {d} [/ math]

[math] H = \ frac {n \, I} {l} [/ math]

creates a magnetic field,
whose strength depends on the current strength, coil length and number of turns.

creates an electrical flow[2]
or stores electrical charge

[math] Q = \ epsilon_0 \, \ epsilon_r \, E \, A [/ math]

[math] n \, \ Phi = n \, \ mu_0 \, \ mu_r \, H \, A [/ math]

creates a magnetic flux

which from the tension
and the capacity depends

[math] Q = C \, U [/ math]

[math] n \, \ Phi = L \, I [/ math]

which of the amperage
and the inductance depends

stores energy in the electric field

[math] \ begin {align} W_ {el} & = \ frac {1} {2} \, Q \, U \ & = \ frac {1} {2} \, C \, U ^ 2 \ & = \ frac {Q ^ 2} {2 \, C} \ \ end {align} [/ math]

[math] \ begin {align} W_ {mag} & = \ frac {1} {2} \, n \ Phi \, I \ & = \ frac {1} {2} \, L \, I ^ 2 \ & = \ frac {(n \ Phi) ^ 2} {2 \, L} \ \ end {align} [/ math]

stores energy[3] in the magnetic field

The capacitance is calculated from the plate area, plate spacing and relative permittivity of the dielectric.

[math] C = \ epsilon_0 \, \ epsilon_r \, \ frac {A} {d} [/ math]

[math] L = n ^ 2 \, \ mu_0 \, \ mu_r \, \ frac {A} {l} [/ math]

The inductance is calculated from the winding area, coil length and relative permeability of the closed Iron core and number of turns.

Calculation of the amount of energy in a coil

The delayed switching on of a lamp with a series-connected coil can be explained by self-induction. Only after the maximum coil current has been reached is all of the energy supplied to the lamp by the power supply unit. Before that, the energy is needed to build up the magnetic field and magnetize the iron core.

The energy stored in the coil can only depend on the strength of the maximum current and properties of the coil, such as the number of turns, etc. In particular, it does not matter how the maximum current is reached, whether quickly or slowly, evenly or unevenly.

To calculate the amount of energy, consider the "discharge" of the coil after it has been disconnected from the voltage source. Before that, a constant current flows, then the coil "discharges" and transfers the energy to a "consumer" like a glow lamp.

The energy transferred to the coil at a time is determined by the power. The total energy is then obtained as an integral over the current rate of change of the energy from the beginning ([math] t = 0 [/ math]) to "much later" ([math] t = \ infty [/ math]):

[math] \ dot W (t) = P (t) [/ math]
[math] W_ {mag} = \ int_0 ^ \ infty \ dot W (t) \, dt = \ int_0 ^ \ infty P (t) \, dt [/ math]

The greater the current strength and the self-induced voltage, the greater the energy flow out of the coil. The power is the product of voltage and current:

[math] W_ {mag} = \ int_0 ^ \ infty P (t) \, dt = \ int_0 ^ \ infty U (t) \, I (t) \, dt [/ math]

The self-induction voltage is proportional to the change in the current strength, the proportionality factor is precisely the inductance of the coil:

[math] U (t) = - L \, \ dot I (t) [/ math]

This gives the integral for the energy of the coil:

[math] W_ {mag} = - L \, \ int_0 ^ \ infty \ dot I (t) \, I (t) \, dt [/ math]

There are different ways to solve this integral. The easiest way is to "guess well". An antiderivative is namely [math] \ frac {1} {2} \, I (t) ^ 2 [/ math], which can be confirmed by deriving it with the chain rule:

[math] \ left (\ frac {1} {2} \, I (t) ^ 2 \ right) ^ {\ cdot} = \ frac {1} {2} \ cdot 2 \, I (t) \ cdot \ dot I (t) [/ math]

So the integral is calculated as follows:

[math] W_ {mag} = - L \, \ int_0 ^ \ infty \ dot I (t) \, I (t) \, dt = L \, \ left [\ frac {1} {2} I (t ) ^ 2 \ right] _0 ^ \ infty = - \ frac {1} {2} \, L \, \ left (I (\ infty) ^ 2 - I (0) ^ 2 \ right) [/ math]

The current strength becomes smaller and smaller over time, in particular [math] I (\ infty) = 0 [/ math]. And [math] I (0) [/ math] is simply the original current strength [math] I [/ math] of the coil:

[math] W_ {mag} = \ frac {1} {2} \, L \, I ^ 2 [/ math]
or with the medium power

For the sake of simplicity, it is assumed that the current intensity falls linearly from [math] I_ {max} [/ math] to 0 within the time [math] t [/ math]. The change in amperage is therefore constant, and thus also the self-induced voltage:

[math] U = - L \, \ dot I = - L \, \ frac {\ Delta I} {\ Delta t} = - L \, \ frac {0-I_ {max}} {0-t} = L \, \ frac {I_ {max}} {t} [/ math]

The power, i.e. the energy output per time, also falls linearly with time, because the voltage is constant and the current strength falls linearly. The mean output therefore corresponds to half the maximum output:

[math] \ bar P = \ frac {1} {2} \, P_ {max} = \ frac {1} {2} \, U \, I_ {max} [/ math]

The amount of energy can now be calculated with the mean power over time:

[math] \ begin {align} W & = \ bar P \, t \ & = \ frac {1} {2} \, U \, I_ {max} \, t \ & = \ frac {1} {2} \, L \, \ frac {I_ {max}} {t} \, I_ {max} \, t \ & = \ frac {1} {2} \, L \ I ^ 2 \ end { align} [/ math]

[math] [/ math]

Footnotes

  1. ↑ To do this, the sample material must be arranged in a closed ring.
  2. ↑ According to Maxwell's equation for electrostatics, the electric flux is just as great as the charge on the plates.
  3. ↑ There are also structural similarities for calculating the kinetic energy of an object of mass m with the velocity v: [math] E_ {kin} = \ frac {1} {2} \, p \, v = \ frac {p ^ 2} {2 \, m} = \ frac {1} {2} \, m \, v ^ 2 [/ math]

Left