# Why are there two terms of both mutal and self inductance

• loesung
In summary, when two inductors are connected in parallel and connected to a battery, the following EMF relations hold: - The EMF in inductor 1 is equal to the negative of the mutual inductance between inductor 1 and 2, multiplied by the rate of change of current in inductor 2. - The EMF in inductor 2 is equal to the negative of the self-inductance of inductor 2, multiplied by the rate of change of current in inductor 2, plus the mutual inductance between inductor 1 and 2, multiplied by the rate of change of current in inductor 1.
loesung

## Homework Statement

Given two inductors, connected in parallel connected to a battery, why do the following emf relations hold?

$$\mathcal{E}_1 = - N_2 A \frac{d B}{d t} = -M\frac{d I_1}{d t } \\ \mathcal{E}_2 = -L_2 \frac{dI_2}{dt}- M \frac{dI_1}{dt}$$

See attachment !

## Homework Equations

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I know for example that the mutual inductance on coil 2 caused by the change in current I_1 is
$$\mathcal{E}_2 = - N_2 A \frac{d B}{d t} = -M\frac{d I_1}{d t },$$

## The Attempt at a Solution

It's not clear to me how there are two terms of the rhs of each of the equations in part (1) !

#### Attachments

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Mutual inductance ## M ## is defined as ## M=\Phi_{1,2}/I_2=\Phi_{2,1}/I_1 ## (the inductance is mutual, i.e. ## M_{1,2}=M_{2,1}=M ##), where ## \Phi_{x,y} ## is the magnetic flux in x due to y. You can then write ## \varepsilon_{1m}=-d \Phi_{1,2}/dt=-M d I_2/dt ## for the effect of change in current in inductor 2 on inductor 1, but you also need to include the self inductance: ## L_1=\Phi_{1,1}/I_1 ## to get another source of EMF in inductor 1: ## \varepsilon_{1s}=-d\Phi_{1,1}/dt=-L_1 dI_1/dt ## . ## \varepsilon_1=\varepsilon_{1m}+\varepsilon_{1s} ##. ... Similarly for inductor 2.

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## Question 1: What is the difference between mutual and self inductance?

Mutual inductance refers to the phenomenon where a changing current in one circuit induces a voltage in another nearby circuit. Self inductance, on the other hand, refers to the tendency of a circuit to resist changes in current flowing through it due to the magnetic field it produces.

## Question 2: Why are there two terms of both mutual and self inductance?

This is because both mutual and self inductance play important roles in understanding and analyzing the behavior of circuits. While mutual inductance is crucial in understanding the interaction between multiple circuits, self inductance is necessary for understanding the behavior of individual circuits.

## Question 3: How are mutual and self inductance related?

Mutual inductance is a result of the interaction between two circuits, where the changing current in one induces a voltage in the other. Self inductance, on the other hand, is a property of a single circuit and is a result of the magnetic field it produces. Both are interconnected and influence each other's behavior.

## Question 4: What factors affect mutual and self inductance?

The main factor affecting mutual inductance is the distance between the two circuits, as a closer proximity will result in a stronger induced voltage. For self inductance, the main factors are the number of turns in the circuit and the presence of a magnetic core material which can increase the magnetic field and therefore, the inductance.

## Question 5: How is the concept of mutual and self inductance applied in real-world applications?

Mutual inductance is used in devices such as transformers, which are crucial in power distribution systems. Self inductance is used in components such as inductors, which are used to store energy in electronic circuits. These concepts are also important in understanding and designing wireless charging systems and electromagnetic sensors.

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