Prof. Penning
18
University of Michigan
Physics 240/Fall
Magnetic Energy & RL Circuits
It requires energy to establish a current in an inductor. Likewise, an inductor carrying a current
stores energy. Let’s say we have an inductor with terminals a and b, and a current i running
through it as shown in Fig. 1. When increasing the current i, an EMF from a to b will be
Figure 1: The symbol for inductor in a circuit diagram
generated, i.e. Vab > 0. But, having voltage and current means we have power!
P = Vab · i
(1)
Meaning we are adding energy to the conductor. Because
Vab = ε = L ·
di
dt
(2)
we rewrite the power as:
P =L·i·
Power is defined as
P =
di
dt
dU
,
dt
i.e. the rate of energy change. So
P =
dU
di
=L·i·
dt
dt
(3)
Or we can express each little differential bit of energy as related to the change in current by
dU = L · i · di
(4)
We want to integrate now:
Z
I
L · i · di
U=
0
where I is the final current to which we increase. Integrating we obtain:
1
U = LI 2
2
(5)
Increasing the current running through an inductor from 0 to I, the energy stored in the inductor
is 12 LI 2 . Once we reach the current I there is no more EMF, the current is not changing, and
no more power and energy transfer. When switching off the source of the current, the inductor
Page 1
Prof. Penning
University of Michigan
Physics 240/Fall
will quickly drain all of its stored energy which generates a negative EMF, Vab < 0. The energy
in the inductor is stored in the magnetic field, similar to how the energy in a capacitor is stored
in the electric field. You might recall, or go back to our discussion of the electric field, that we
discovered the energy density in an electric field this way:
1
U = ε0 E 2
2
We can do something similar for a magnetic field using a toroid, shown in Fig. 2. The magnetic
Figure 2: Dimensions of our toroid [1]
field inside is obtained from Ampere’s law:
µ0 N i
2πr
With Φ = B · A we calculate the inductance L using the expressoin for self-inductance:
B2πr = µN I ⇒ B =
L=
N ·Φ
µ0 · N 2 · A
=
i
2πr
and the energy stored:
1
U = LI 2
2
the energy per unit volume is:
U
1
u∼
=
2πrA
2
µ0 N 2 A
1
· I2 ·
2πr
2πrA

2
1 N 2I 2
1 1  µ0 N I 
= µ0
=


2
2 (2πr)
2 µ0 | 2πr
{z }
B
We’re using the approximate sign because this relationship holds only if A is small relative to r.
So we have
1 2
u=
B
(6)
2µ0
Page 2
Prof. Penning
University of Michigan
Physics 240/Fall
It turns out this relation is true for all types of magnetic fields, independently if we discuss toroids,
solenoids, fields from a wire, etc. If the field is in some material with magnetic permeability, then
the expression modifies simply to:
1
u=
(7)
B2
2µr µ0
As one would expect.
18.1
R-L-Circuits
Let’s study the circuits to understand better how inductors in a circuit behave. Figure 3 shows
a circuit consisting of a resistor (R) an inductor (L), a power source, and a switch connected in
series. This is called an R-L circuit. The inductor’s function is to prevent rapid rises in current.
Figure 3: A R-L circuit
Imagine a time t = 0 when we close the switch. If i is the current at time t, then
Vab = i · R,
Vbc = L ·
di
dt
Using Kirchoff’s loop rule we know:
ε−i·R−L
di
=0
dt
Rearranging we get:
ε−i·R
di
=
dt
L
This is a first-order differential equation, similar to what we’ve seen when discussing the RC
circuit. Because at t = 0 there is no current we simplify:
di
ε
=
dt initial L
Page 3
Prof. Penning
University of Michigan
Physics 240/Fall
This shows us that the greater the inductance the slower the change. Remember we said that
inductance plays somewhat the role of mass in mechanics. As the current increases the term i·R
L
slows the rate of increase until we finally reach the maximum current I:
ε−I ·R
ε
di
=0=
⇒I=
dt final
L
R
Now the inductor no longer provides any resistance. What happens between the initial and final
states? We can answer this by rearranging the expression again:
ε − iR
di
ε/R − i
di
dt
di
=
⇒
=
⇒−
=
L
dt
L/R
dt
L/R
i − ε/R
so we get
ε−i·R
di
R
di
=
⇒
= − dt
dt
L
i − ε/R
L
Now the current is on the left and the time on the right. We now integrate the expressions. For
aesthetic reasons I want my final variables to be i and t, so I rename my integration variables to
i′ and t′
Z i
Z t
di′
R ′
i − ε/R
R
=−
dt ⇒ ln
=− t
′
−ε/R
L
0 i − ε/R
0 L
Solving for i we obtain an expression for the current over time while charging:
ε −R
·t
L
1−e
i=
(8)
R
Note the expression in the exponent. We call
τ=
L
R
time constant for R-L circuits
(9)
Figure 4 shows the change over time between the initial and final state. The current starts at
zero, and as t → ∞, I(t) approaches ε/R asymptotically. The induced EMF V (t) is directly
proportional to dI/dt, or the slope of the curve. Hence, while at its greatest immediately after
the switches are thrown, the induced EMF decreases to zero with time as the current approaches
its final value of ε/R. The circuit then becomes equivalent to a resistor connected across a source
of emf [2]. At the value of the time constant, the inductor about 63% charged:
1
≈ 63% of I
i=I · 1−
e
See Fig. 4 for the corresponding graphs for current i and emf.
Similar when the inductor is fully charged and we discharge it by bypassing the original power
source. In that case, the instantaneous current is described by:
i = I0 e−(R/L)t
We can obtain this by retracing our steps from above for the discharging case. Now the time
constant τ = L/R is the time it takes for the current to decrease to 1/e ≈ 37% of its original
value. The induced emf when discharging the inductor can be very large, larger than the initial
charging voltage. If for example, the resistor is a light bulb it can possibly burn out when
discharging.
Page 4
Prof. Penning
University of Michigan
Physics 240/Fall
Figure 4: Time variation of (a) the electric current and (b) the magnitude of the induced voltage
across the coil of a series R-L circuit [2] during charging.
18.2
Magnetic Materials
We discussed in the last few lectures how to create magnetic fields from currents. What about
permanent magnets? What happens when we magnetize an object?
Any material is made of atoms, which themselves are made of a nucleus (protons & neutrons) and
electrons orbiting the nucleus. The nuclei are positively charged and the electrons are negatively
charged. The electrons whizzing around the nuclei act like a little current loop, they produce a
Figure 5: Hydrogen atom consisting of one electron and one proton (nucleus)
magnetic field and interact with other magnetic fields. Whenever negatively charged electrons
are unevenly distributed around nuclei, or in molecules whenever electrons are unevenly shared
between two atoms in a covalent bond a dipole is present. Dipole simply means ‘two poles’, two
electrical charges, one negative and one positive. The magnetic dipole moment is a measure of the
strength of the magnetic field produced by the orbital angular momentum of an electron [2]. In
most materials the little magnets those electrons cause are not aligned, resulting in a net magnetic
field of zero. But when they are (approximately) aligned the material becomes magnetic, Fig. 6.
In a permanent magnet, these atoms/molecules are aligned and somewhat locked. We call such
materials ferromagnetic. What happens when those dipoles are allowed to move and we apply
an external magnetic field? Then due to
⃗
F⃗ = q · ⃗v × B
Page 5
Prof. Penning
University of Michigan
Physics 240/Fall
Figure 6: Magnetic molecule alignment in metal
the loops experience a torque such that the magnetic fields created by them align with the external
field. A material that can be magnetized is called ’paramagnetic’. Examples are aluminum,
sodium, and tungsten.
In some materials the induced magnetic field points in the opposite direction of the external
fields. These are called diamagnetic. Examples are bismuth and carbon. Our ‘little loop’ analogy
does not explain diamagnetism for which we need quantum mechanics to explain. In fact, all
materials are diamagnetic, but the diamagnetic contribution is often much weaker and therefore
are negligible compared to the ferromagnetic or paramagnetic contributions. Hence few materials
appear diamagnetic.
By placing a ferro- or paramagnetic material in a magnetic field we can enhance its strength similar
to a dielectric in a capacitor leads to the greater charge stored and therefore greater capacitance.
The permeability is the measure of magnetization that a material obtains in response to an applied
magnetic field. Permeability is represented by µ.
Example: We can illustrate the behaviors using the solenoid coil we already discussed. The
Figure 7: A solenoid along [3]
solenoid has n turns per length and current I the magnetic field within the core is:
⃗ 0 = µ0 · I · n
B
(10)
Filling the solenoid with a linear magnetic material with permeability µr , then the field is enhanced
by:
⃗ = µr µ0 · I · n = µr · B
⃗0
B
(11)
⃗ 0 field without material and then multiply
Hence for any structure, we can simply calculate the B
by µr if the material is linear to obtain the enhanced B-field. Some examples of ferromagnetic
Page 6
Prof. Penning
University of Michigan
Physics 240/Fall
materials are given in Table 1. The permeability depends strongly on environmental factors such
Material
Ferrite
Stainless Steel
Nickel
ur
350-500
1000-1800
100
Table 1: Permeability of some ferromagnetic materials
as temperature, field strengths, etc. Paramagnetic materials usually have much smaller µr ∼ 1.
Page 7
Prof. Penning
University of Michigan
Physics 240/Fall
References
[1] “Toroidal Magnetic Field — hyperphysics.phy-astr.gsu.edu,” http://hyperphysics.phy
-astr.gsu.edu/hbase/magnetic/toroid.html, [Accessed 27-Dec-2022].
[2] “University Physics II - Thermodynamics, Electricity, and Magnetism (OpenStax),” (2022),
[Online; accessed 2022-12-26].
[3] S. Bhuyan, “Solenoid Magnetic Field: Definition and Equation — sciencefacts.net,” https:
//www.sciencefacts.net/solenoid-magnetic-field.html, [Accessed 24-Dec-2022].
Page 8