Energy stored in coil with given relationship between current and flux

In summary, the conversation is about determining the energy stored in a coil when the current varies from 0 to I, using the relationship between the current and flux given by \Phi = \frac{ai}{b+ci}. The attempt at a solution involves differentiating the relationship and setting up two integrals, which may require integration by parts.
  • #1
Trip1
4
0

Homework Statement



The relationship between the current in an N-turn coil and the flux created by it is given as:

[tex] \Phi = \frac{ai}{b+ci}[/tex]

Determine the energy stored in the coil when the current varies from 0 to I.

Homework Equations



[tex] W = N \int{i d\Phi} [/tex]

The Attempt at a Solution



Started by differentiating the given relationship with respect to i, using the quotient rule

[tex] \frac{d\Phi}{di} = \frac{a(b+ci) - cai}{(b+c{i}^2)} \\ \\
\Rightarrow d\Phi = \frac{a(b+ci)}{{(b+ci}^2)}di - \frac{aci}{(b+ci)^2}di

[/tex]

I then proceed to substitute this expression for [tex]d\Phi[/tex] into the equation for W above, and setup two integrals (one for each term), integrating with respect to i from 0 to I.

Problem is, the integrations are very complex to do by hand, and they aren't in a general form to lookup in a table. This leads me to believe I've made a mistake somewhere, but i can't seem to find it.

Any help would be greatly appreciated, thanks.
 
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  • #2
Trip1 said:

Homework Equations



[tex] W = N \int{i d\Phi} [/tex]
.

You might try integration by parts.
 

What is the relationship between current and flux in a coil?

The relationship between current and flux in a coil is described by Faraday's law of induction, which states that the induced electromotive force (EMF) in a closed loop is equal to the negative rate of change of magnetic flux through the loop. In other words, the current flowing through a coil is directly proportional to the rate of change of magnetic flux passing through the coil.

How is the energy stored in a coil with a given relationship between current and flux?

The energy stored in a coil with a given relationship between current and flux is known as magnetic energy. This energy is stored in the form of a magnetic field, which is created by the current flowing through the coil. The amount of energy stored in a coil depends on the number of turns in the coil, the current flowing through it, and the permeability of the core material.

What factors affect the energy stored in a coil?

The energy stored in a coil is affected by several factors, including the number of turns in the coil, the current flowing through it, and the permeability of the core material. It is also influenced by the shape and size of the coil, the type of core material used, and the strength of the magnetic field produced by the coil.

How can the energy stored in a coil be calculated?

The energy stored in a coil can be calculated using the formula E = 1/2 LI^2, where E is the energy stored, L is the inductance of the coil, and I is the current flowing through the coil. This formula applies to ideal coils without any resistance or losses. In real-world applications, the calculation may be more complex and require considerations for factors such as resistance and non-ideal characteristics of the coil.

How is the energy stored in a coil used in practical applications?

The energy stored in a coil has many practical applications, such as in transformers, motors, and generators. In transformers, it is used to transfer electrical energy from one circuit to another through mutual induction. In motors and generators, it is used to convert electrical energy into mechanical energy and vice versa. The energy stored in a coil is also used in a variety of electronic devices, such as inductors and chokes, to control the flow of current and filter out unwanted signals.

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