Subject Code PHY 3 Physics 3

Module Code 10.0 Inductors

Lesson Code 10.3 Mutual - Inductance
Time Frame 30 minutes

TA1 ATA2
Components Tasks
(min) (min)
Target By the end of this learning guide, you should be able to: 1

● articulate the physical meanings of and differentiate between

self- and mutual- inductance; and
● calculate for mutual-inductance.
Hook In the previous lesson, we learned self-inductance in which we 1
consider an isolated circuit wherein a current that is present in the
circuit sets up a magnetic field that causes a magnetic flux through the
same circuit; this flux changes when the current changes. Now, we
consider two neighboring coils of wire, as in Figure 1. A current
flowing in coil 1 produces a magnetic field B and hence a magnetic
flux through coil 2 changes as well. According to Faraday’s law, this
induces the electromotive force (emf) in coil 2. In this way, a change
in the current in one circuit can induce a current in a second circuit.
This process is described in terms of mutual inductance which we will
discuss in this learning guide.

Young & Freedman

Figure 1: A current i1 in coil 1 gives rise to a magnetic flux through coil 2.

If i1 changes, an emf is induced in coil 2; this is described in terms of mutual
inductance.

Ignite We learned that an inductor generates an induced emf within itself as 15

a result of the changing magnetic field around its own turns. When this
emf is induced in the same circuit in which the current is changing this
effect is called self-induction. However, when the emf is induced into
an adjacent coil situated within the same magnetic field, the emf is said
to be induced magnetically, inductively or by mutual induction (Figure

1
Time allocation suggested by the teacher.
2
Actual time allocation spent by the student (for information purposes only).

Physics 3 Mutual – inductance Page 1 of 8

 2). Then when two or more coils are magnetically linked together by a
common magnetic flux, they are said to have the property of mutual
inductance (M). Mutual Inductance is the interaction of one’s coil
magnetic field on another coil as it induces an emf (Ɛ) in the adjacent
coil.

OpenStax

Figure 2: These coils can induce emfs Ɛ in one another. Their mutual inductance M
indicates the effectiveness of the coupling between them. Here a change in current i 1
in coil 1 is seen to induce an emf Ɛ2 in coil 2. (Note that “E2 induced" represents the
induced emf Ɛ2 in coil 2.)

The amount of mutual inductance that relates one coil to another

depends very much on the relative positioning of the two coils. If one
coil is positioned next to the other coil so that their physical distance
apart is small, then nearly all of the magnetic flux generated by the first
coil will interact with the second coil inducing a relatively large emf
and therefore producing a large mutual inductance value.
Likewise, if the two coils are farther apart from each other or at
different angles, the amount of induced magnetic flux from the first
coil into the second will be weaker, producing a much smaller induced
emf and therefore a much smaller mutual inductance value. So, the
effect of mutual inductance largely depends upon the relative positions
or spacing of the two coils as demonstrated in Figure 3. Further, the
mutual inductance that exists between the two coils can be greatly
increased by positioning them on a common soft iron core or by
increasing the number of turns of either coil as would be found in a
transformer.

Physics 3 Mutual – inductance Page 2 of 8

 ElectronicsTutorials

Figure 3: The mutual inductance that exists between the two coils can be greatly
affected by their distance relative with each other.

Let’s analyze the situation mathematically. When i1 changes, ΦB2

changes; this changing flux induces an emf Ɛ2 in coil 2, given by

𝛥𝛷𝐵2
Ɛ2 = −𝑁2 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 1
𝛥𝑡

To represent the proportionality of ΦB2 and i1 in the form

ΦB2 = (constant)i= i1, but instead it is more convenient to include the
number of turns N2 in the relation. Introducing a proportionality
constant M21, called the mutual inductance of the two coils, we write

𝑁2 𝛷𝐵2 = 𝑀21 𝑖1 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 2

where ΦB2is the flux through a single turn of coil 2. From this,

𝑁2 𝛥𝛷𝐵2 𝛥𝑖
= 𝑀21 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 3
𝛥𝑡 𝛥𝑡

and we can rewrite equation 2 as

𝑁2 𝛷𝐵2
𝑀21 = 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 4
𝑖1

For the opposite case in which a changing current i2 causes a changing

flux ΦB1 and an emf Ɛ1 in coil 1. We might expect that the
corresponding constant M12 would be different from M21 because in
general the two coils are not identical and the flux through them is not
the same. It turns out, however, that M12 is always equal to M21, even
when the two coils are not symmetric. This is called the common value
simply the mutual inductance, denoted by the symbol M without
subscripts; it characterizes completely the induced-emf interaction of
two coils. Then, we can write

Physics 3 Mutual – inductance Page 3 of 8

 𝛥𝑖1 𝛥𝑖2
Ɛ2 = −𝑀 𝑎𝑛𝑑 Ɛ1 = −𝑀 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 5 & 6
𝛥𝑡 𝛥𝑡

where the mutual inductance M is

𝑁2 𝛷𝐵2 𝑁1 𝛷𝐵1
𝑀= = 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 7
𝑖1 𝑖2

The negative signs in equations 5 and 6 are a reflection of Lenz’s law.

The first equation presents that a change in current in coil 1 causes a
change in flux through coil 2, inducing an emf in coil 2 that opposes
the flux change. In the second equation, the roles of the two coils are
interchanged.

The SI unit of mutual inductance is called the henry (1 H), in honor of

the American physicist Joseph Henry (1797 – 1878). From equation 7,
one henry is equal to one weber per ampere. Other equivalent units
obtained by using equations 5 and 6, one volt-second per ampere, one
ohm-second, or one joule per ampere squared:

𝑊𝑏 𝑠 𝐽
1𝐻 =1 = 1 𝑉. = 1𝛺. 𝑠 = 1 2
𝐴 𝐴 𝐴

Now, let’s calculate for mutual inductance

Example 1: A current in one coil changes uniformly from 2.0 A to 0.50

A in 45 ms. An emf whose magnitude is 5.5 V is seen to develop in a
second coil located close to the first. What is the mutual inductance of
the two coils? (Giancoli, 2003)

Solution:

The magnitude of the emf is given by equation 5 without the minus

sign. Therefore, the mutual inductance is

𝛥𝑖1
Ɛ2 = −𝑀
𝛥𝑡

𝛥𝑡 (45𝑥10−3 𝑠)
𝑀 = Ɛ2 = 5.5 𝑉 = 0.17 𝐻
𝛥𝑖1 1.5𝐴

Example 2: In one form of Tesla coil (a high-voltage generator that

you may have seen in a science museum), a long solenoid with length
0.50 m and cross section area of 10 cm2 is closely wound with 1000
turns of wire. A second coil of 10 turns surrounds it at its center. Find
the mutual inductance. (Young and Freedman, 1999)

Solution

Physics 3 Mutual – inductance Page 4 of 8

 A current i1 in the solenoid sets up a magnetic field B1 at its center, the
magnitude of B1 is

𝜇𝑜 𝑛1 𝑖1
𝐵1 =
𝑙

The flux ΦB through a cross section of the solenoid equals BA. Since a
very long solenoid produces no magnetic field outside of its coil, this
is equal to the flux ΦB2 through each turn of the outer, surrounding
coil, no matter what the cross-section area of the outer coil. From
equation 7

𝑁2 𝛷𝐵2 𝑁2 𝐵1 𝐴 𝑁2 𝜇𝑜 𝑁1 𝑖1 𝜇𝑜 𝐴𝑁1 𝑁2
𝑀= = = 𝐴=
𝑖1 𝑖1 𝑖1 𝑙 𝑙

Substituting the values

𝑊𝑏
(4𝜋 𝑥 10−7 ) (1.0 𝑥10−3 𝑚2 )(1000)(10)
𝑀= 𝐴. 𝑚
0.50 𝑚

𝑊𝑏
𝑀 = 25 𝑥 10−6 = 25 𝑥 10−6 𝐻 = 25𝜇𝐻
𝐴

Example 3: A circuit consists of a primary coil and a secondary coil,

each with the same number of turns, wrapped around an iron core and
they are near each other. The coils have a mutual inductance of 50 mH.
A current in the primary coil increases the magnetic flux through the
core by 6.48 mWb. The current induced in the secondary coil is 1.6 A.
How many turns does the coil have?

Solution

From equation 7, we derived the formula for N

𝑁1 𝛷𝐵1
𝑀=
𝑖2

Since N1 = N2

𝑀𝑖1
𝑁1 = 𝑁2 =
𝛷𝐵2

Substituting the values

(50 𝑥 10−3 𝐻)(1.6 𝐴)

𝑁1 = 𝑁2 = = 12
6.48 𝑥 10−3 𝑊𝑏

Physics 3 Mutual – inductance Page 5 of 8

 Navigate Direction: Solve the following problems on a sheet of paper. Follow 10
your teacher’s instruction on how to submit your answers.

1. A long thin solenoid of length l cross sectional area A contains N1

closely packed turns of wire. Wrapped tightly around it is an
insulated coil of N2 turns. Assume all the flux from coil 1 passes
through coil 2, calculate the mutual inductance.
(Giancoli, 2003)

2. A 30-cm long coil with 1, 500 loops is wound on an iron core (μ =

3000 μo) along with a second coil of 800 loops. The loops of each have
a radius of 2.0 cm. If the current in the first coil drops uniformly from
3.0 A to zero in 8.0 ms, determine (a) the emf induced in the second
coil, and (b) the mutual inductance. (Giancoli, 2003)
Knot In summary, we learned that 3

● Mutual Inductance is the interaction of one’s coil magnetic

field on another coil as it induces an emf (Ɛ) in the adjacent
coil.

● The mutual inductance M is given by the equation

𝑁2 𝛷𝐵2 𝑁1 𝛷𝐵1
𝑀= =
𝑖1 𝑖2

References:

1. Cutnell, J., & Johnson, K. (2012). Physics (9th ed.). USA: John Wiley & Sons, Inc.
2. Giancoli, D. (2003). Physics (5th ed.). USA: Prentice Hall, Inc.
3. ElectronicsTutorials. (2019). The Inductor. Retrieved from https://www.electronics-
tutorials.ws/inductor/inductor.html
4. Mini Physics. (n.d.). Self – inductance and Inductors. Retrieved from
https://www.miniphysics.com/uy1-self-inductance-inductors.html
5. Openstax. (2020). Inductance. Retrieved from https://openstax.org/books/college-
physics/pages/23-9-inductance
6. Openstax. (2020). Self – inductance and inductors. Retrieved from
https://openstax.org/books/university-physics-volume-2/pages/14-2-self-inductance-and-
inductors
7. Young, H. and Freedman, R. (1999 & 2008). Sears and Zemansky’s University Physics with
Modern Physics (10th & 12th eds.). San Francisco: Pearson Education

Prepared by: Sheryl A. Salvador Reviewed by: Jericho V. Narvasa

Position: Special Science Teacher IV Position: Special Science Teacher I

Campus: Ilocos Region Campus Campus: Ilocos Region Campus

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