CHAPTER 5
Electromagnetic Induction
5.1. Electromotive Force and Magnetic Flux
5.1.1 Electromotive force
The electromotive force (e.m.f.) is defined as the work done per unit charge to move the charge in a closed
circuit loop. Thus, if we have a charge q, then by definition, we have
˛
1
E= ⃗F • d⃗l (5.1.1)
q L
where E is the emf. The SI unit of this quantity is Volts. In Chapter 1, we had defined the electric
potential difference as the work expended per unit charge, so the emf is simply a very specific type of
⃗
electric potential. Since we can define the electric field as ⃗E = q
F
, we can rewrite the above equation
as ˛
E= ⃗E • d⃗l (5.1.2)
L
Unfortunately, the "force" in the name tends to confuse students when they first learn of this term.
The emf is not a force at all but an expression of the energy required (per charge) to drive a bunch
of charges around inside a conductor to create a current. The misleading name has stuck around
primarily due to historical reasons; these days, modern convention is trying to replace emf with the
more accurate term source voltage. However, to maintain consistency with the way most textbooks
cover this quantity, we will continue to use the term emf in this document.
Historically, when a source of electric current (such as a battery) was attached to a wire, it was thought
that the source exerted a force on the electrons causing them to move inside the conductor (hence the
term electromotive) and create a current. The higher the emf, the higher the current produced. With
our modern understanding of the subject, we now know that the source creates an electric potential
difference which in turn exerts a force on the charge carriers (i.e. electrons) to make them move.
To create a higher potential difference, we need to expend more electrical/magnetic energy. From a
mathematical perspective, there is no difference between emf and electric potential.
The term emf is used for the potential generated by devices that are a source of electric current or electric
potential. Such devices are called transducers and they convert other forms of energy into electrical
energy. Obvious examples are batteries (chemical to electric) and generators (mechanical to electric).
103
, CHAPTER 5. ELECTROMAGNETIC INDUCTION 104
5.1.2 Magnetic flux
Let us now imagine some surface that is bounded by a closed curve having N loops and immersed
in some external magnetic field B.⃗ We define the magnetic flux as the product of the total magnetic field
moving through the cross-sectional area perpendicular to the surface. Mathematically
ˆ
Φ=N ⃗ • dS
B ⃗ (5.1.3)
S
The SI unit of magnetic flux is called a Weber (Wb) and is defined as 1 Wb = 1 T.m2 . It is not uncom-
mon for some textbooks to specify the units of the magnetic field as Wb/m2 instead of using Tesla.
For simple geometric shapes (circles, squares etc.), we can write the flux as
Φ = NBA cos θ (5.1.4)
where A is the area of the coil and θ is the angle between the magnetic field and the surface area
vector.
Relation between the magnetic flux and the emf
Let us imagine some closed loop is moving in space through a constant magnetic field B. ⃗ This means
that the flux is no longer constant, since the interaction between the area of the loop and the magnetic
field inside the integral changes with time. Additionally, let us imagine there are charged particles
in the loop. The motion of the loop will generate a magnetic force via the Lorentz force law. From
equation 5.1.1, this force will induce (create) an emf inside the loop which we will denote as E.
Theorem: For a magnetic field that is constant in time, the induced emf E around
a loop of any shape moving in any manner is related to the magnetic flux through
the loop by
dΦ
E=− (5.1.5)
dt
Proof With Figure1 5.1.1 as a reference, let us
imagine a surface S that is bounded by a closed
boundary loop (with N = 1). This loop occupies
the curve C1 at some time t and then moves to
the curve C2 after some time at t + dt with a ve-
locity ⃗v. We imagine the loop is immersed in a
⃗ So the original flux
magnetic field given by B.
through the loop at time t is given by
ˆ
Φ(t) = B⃗ • dS
⃗
S
and at the instant t + dt , the flux changes to
ˆ
Φ(t + dt) = ⃗ • dS
B ⃗
ˆS+dS
ˆ
= ⃗ ⃗
B • dS + ⃗ • dS
B ⃗ F IGURE 5.1.1: Setup for the proof
S dS
1
Image taken from Purcell, E.M. & Morin, D.J. (2013) Electricity and magnetism (3rd ed.). Cambridge University Press.
Page 352.
Electromagnetic Induction
5.1. Electromotive Force and Magnetic Flux
5.1.1 Electromotive force
The electromotive force (e.m.f.) is defined as the work done per unit charge to move the charge in a closed
circuit loop. Thus, if we have a charge q, then by definition, we have
˛
1
E= ⃗F • d⃗l (5.1.1)
q L
where E is the emf. The SI unit of this quantity is Volts. In Chapter 1, we had defined the electric
potential difference as the work expended per unit charge, so the emf is simply a very specific type of
⃗
electric potential. Since we can define the electric field as ⃗E = q
F
, we can rewrite the above equation
as ˛
E= ⃗E • d⃗l (5.1.2)
L
Unfortunately, the "force" in the name tends to confuse students when they first learn of this term.
The emf is not a force at all but an expression of the energy required (per charge) to drive a bunch
of charges around inside a conductor to create a current. The misleading name has stuck around
primarily due to historical reasons; these days, modern convention is trying to replace emf with the
more accurate term source voltage. However, to maintain consistency with the way most textbooks
cover this quantity, we will continue to use the term emf in this document.
Historically, when a source of electric current (such as a battery) was attached to a wire, it was thought
that the source exerted a force on the electrons causing them to move inside the conductor (hence the
term electromotive) and create a current. The higher the emf, the higher the current produced. With
our modern understanding of the subject, we now know that the source creates an electric potential
difference which in turn exerts a force on the charge carriers (i.e. electrons) to make them move.
To create a higher potential difference, we need to expend more electrical/magnetic energy. From a
mathematical perspective, there is no difference between emf and electric potential.
The term emf is used for the potential generated by devices that are a source of electric current or electric
potential. Such devices are called transducers and they convert other forms of energy into electrical
energy. Obvious examples are batteries (chemical to electric) and generators (mechanical to electric).
103
, CHAPTER 5. ELECTROMAGNETIC INDUCTION 104
5.1.2 Magnetic flux
Let us now imagine some surface that is bounded by a closed curve having N loops and immersed
in some external magnetic field B.⃗ We define the magnetic flux as the product of the total magnetic field
moving through the cross-sectional area perpendicular to the surface. Mathematically
ˆ
Φ=N ⃗ • dS
B ⃗ (5.1.3)
S
The SI unit of magnetic flux is called a Weber (Wb) and is defined as 1 Wb = 1 T.m2 . It is not uncom-
mon for some textbooks to specify the units of the magnetic field as Wb/m2 instead of using Tesla.
For simple geometric shapes (circles, squares etc.), we can write the flux as
Φ = NBA cos θ (5.1.4)
where A is the area of the coil and θ is the angle between the magnetic field and the surface area
vector.
Relation between the magnetic flux and the emf
Let us imagine some closed loop is moving in space through a constant magnetic field B. ⃗ This means
that the flux is no longer constant, since the interaction between the area of the loop and the magnetic
field inside the integral changes with time. Additionally, let us imagine there are charged particles
in the loop. The motion of the loop will generate a magnetic force via the Lorentz force law. From
equation 5.1.1, this force will induce (create) an emf inside the loop which we will denote as E.
Theorem: For a magnetic field that is constant in time, the induced emf E around
a loop of any shape moving in any manner is related to the magnetic flux through
the loop by
dΦ
E=− (5.1.5)
dt
Proof With Figure1 5.1.1 as a reference, let us
imagine a surface S that is bounded by a closed
boundary loop (with N = 1). This loop occupies
the curve C1 at some time t and then moves to
the curve C2 after some time at t + dt with a ve-
locity ⃗v. We imagine the loop is immersed in a
⃗ So the original flux
magnetic field given by B.
through the loop at time t is given by
ˆ
Φ(t) = B⃗ • dS
⃗
S
and at the instant t + dt , the flux changes to
ˆ
Φ(t + dt) = ⃗ • dS
B ⃗
ˆS+dS
ˆ
= ⃗ ⃗
B • dS + ⃗ • dS
B ⃗ F IGURE 5.1.1: Setup for the proof
S dS
1
Image taken from Purcell, E.M. & Morin, D.J. (2013) Electricity and magnetism (3rd ed.). Cambridge University Press.
Page 352.
