# Inductance and Coils Homework Solution

• 6Stang7
In summary, the two coils, Coil 1 and Coil 2, have different inductances and number of turns. They are rigidly positioned with respect to each other and have a mutual inductance of 3.0mH. A 6.0mA current in Coil 1 is changing at 4.0 A/s. The magnetic flux that links Coil 1 to Coil 2 is a combination of the mutual inductance (M) and the self inductance of Coil 1 (L1) while the flux that links Coil 2 to Coil 1 is a combination of the mutual inductance (M) and the self inductance of Coil 2 (L2). The induced
6Stang7

## Homework Statement

Coil 1 has L=25mH and N=100 turns. Coil 2 has L=40mH and N=200. The coils are rigidly positioned with respect to each other, and their mutual inductance is 3.0mH. A 6.0mA current in coil 1 is changing at 4.0 A/s. (a)What magnetic flux Phi12 links coil 1 to coil 2, and what is the self-induced emf that appears in coil 1? (b) What magnetic flux Phi21 links coil 2 to coil 1, and what is the mutually induced emf that appears in coild 2?

## Homework Equations

I know I need E=-L(dI/dt) for the self induced emf and E=-M(dI/dt) for the emf induced in coil2.

## The Attempt at a Solution

Solving for both emfs easy and I understand it. What I don't get is what is ment by the magnetic flux that links coil 1 to coil 2 and visea-versa. Would Phi12=L1I1+MI2 and Phi21=L2I2+MI1?
Given that I don't have length (but since I do have number of total turns and would assume that I can assume a length of 1m?) or cross-sectional area, I don't see how you could ever calculate the induced current. I think I have a conceptual problem here and my book is sort of famous for this. Anyone?

There is always something confusing about "the flux through a coil".
We know that a closed loop spans a surface, and this is not different for a coil. However, this is a "multi-layered" surface. The flux through this "multi-layered" surface will be approximately N times the flux through a simple surface spanning the coil "only once", when N is the number of windings.
The simple surface would be a disk, and the "multi-layered" surface would be some kind of helicoidal surface with a very small step (the step of the windings).

If we call $$\phi_{multi}$$ the correct, multi-layered surface attached to the coil, then we have that the EMF is the time variation of this flux. However, we usually call the flux through a coil, the flux through the simple surface $$\phi_{simple}$$. So we have:

$$EMF = \frac{d \phi_{multi}}{dt} = N \frac{d \phi_{simple}}{dt}$$

We also know that $$EMF_1 = L \frac{d i_1}{dt} + M \frac{d i_2}{dt}$$

From this, it follows that
$$\phi_{simple} = \frac{L i_1 + M i_2}{N}$$

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why do you have emf=-M(dI2/dt)? isn't it emf=-M(dI1/dt)? And what is N in this case? If I am lokking for the flux of 1 onto 2, then would I set N to be coil 1's turns or coil 2? I don't know the current in coil 2 either; is there a way to solve for that?

I guess what I am not getting here is the conceptual difference between the two fluxes.

$$L i_1$$ is the flux seen by the (multilayered) surface on coil 1 by the current in coil 1. $$M i_2$$ is the flux seen by the multilayered surface on coil 1 by the current in coil 2.
Now, we usually call the flux, not the flux through the multilayered surface, but by the simple surface, which is then N times less, with N the number of windings of coil 1.

EDIT: you can put in minus signs. That depends on some conventions.

EDIT 2: I was taking coil 1 here as an abstract example. Of course something similar can be written about coil 2.

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so for this case, the flux that links coil 1 to coil 2 is MI2 + LI1 and the flux that links coil 2 to coil 1 is LI2 + MI1.

now, is there any way to solve for the induced current in coil 2? Wouldn't I need more information such as length and area of coil 2?

anyone got an idea?

6Stang7 said:
so for this case, the flux that links coil 1 to coil 2 is MI2 + LI1 and the flux that links coil 2 to coil 1 is LI2 + MI1.

No. The flux (on the helicoidal surface) induced in coil 1 by the current in coil 2 is M I2.
The flux induced in coil 1 by the current in coil 1, is L1 I1.
The total flux is the sum of both contributions.

And if you want the flux in a single "disk" surface, you have to divide by the number of windings of coil 1.

In the same way:

The flux (on the helicoidal surface) induced in coil 2 by the current in coil 1 is M I1.
The flux induced in coil 2 by the current in coil 2, is L2 I2.
The total flux is the sum of both contributions.

And if you want the flux in a single "disk" surface, you have to divide by the number of windings of coil 2.

now, is there any way to solve for the induced current in coil 2? Wouldn't I need more information such as length and area of coil 2?

No: obviously, what you need are the currents (you have them), the coefficients of induction (L1, L2 and M) and the number of windings. All the geometry is already included in the coefficients of induction.

## 1. What is inductance?

Inductance is a property of an electrical circuit that describes the ability of a circuit to generate an electromotive force (EMF) in response to a change in current. It is measured in units of Henry (H).

## 2. How does inductance affect an electrical circuit?

Inductance affects an electrical circuit by creating a back EMF that opposes the change in current. This can cause delays in the circuit’s response and can also store energy which can be released later. It can also cause voltage spikes and can be used to filter out unwanted frequencies.

## 3. What is a coil and how does it relate to inductance?

A coil is a structure made up of conductive material, most commonly copper wire, wrapped around a core. The number of turns, the shape of the coil, and the material used all affect the inductance of the coil. A coil can be used to increase inductance in a circuit.

## 4. How can I calculate the inductance of a coil?

The inductance of a coil can be calculated using the formula L=N^2μ0μrA/l, where N is the number of turns, μ0 is the permeability of free space, μr is the relative permeability of the core material, A is the cross-sectional area of the coil, and l is the length of the coil. Alternatively, you can use an inductance meter to measure the inductance directly.

## 5. What are some practical applications of inductance and coils?

Inductance and coils are used in a wide range of electronic devices and circuits, such as transformers, motors, generators, speakers, and antennas. They are also used in power supplies, filters, and oscillators. In addition, they play a crucial role in wireless charging, wireless power transfer, and inductive coupling for communication between electronic devices.

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