Electricity (7/11) – Circuits, Resistors, Inductors, and Capacitors

Inductors and Inductance (V = LdI/dt = Ldi/dt)

• note 1: This section is based on the excellent Khanacademy videos that I list at the end of this section. 
• note 2: I’ve mostly used the lower case i for current in this section. It could have been written with the uppercase I as well.

Inductor

Picture: Inductor Circuit Symbol

Self Inductance

Consider the following simple circuit.

Picture: Simple Circuit with Inductor (Coil)

The circuit contains a coil wound around a core of iron which we can call an inductor element (we could call it a Solenoid as well).

When the switch is closed in this circuit, current flows and a magnetic field is generated.

The bulb will take a little time to reach full brightness.

When it reaches maximum brightness, we can use Ohms Law to define the Voltage, Current, Resistance relationship (V = iR = IR).

When the switch is closed making the circuit above “live”, the following occurs:

• The current starts increasing, Δi, 
• the magnetic field B in the coil starts increasing ,ΔB (by Ampère’s  Circuital Law) ,
• and so does the magnetic flux Φ_B  in the coil,  ΔΦ_B

Faraday’s law tells us what is happening in the coil (we’ll write it in differential notation indicating an instantaneous infinitesimal change):

(1) EMF=−NdΦ_B/dt

An EMF is induced , and ,according to Lenz’s Law, it opposes the current.

For a coil or conductor the magnetic flux is proportional to the current,

Φ_B ∝ i

Lets equate them through a proportionality constant L (and we’ll call it the Self Inductance).

NΦ_B  = Li (we’ll look at this again in equation (3) below)

Rearrange Faraday’s Law Equation (1) and substitute for NΦ_B

EMF= −d/dt(NΦ_B) = −d(Li)/dt 

EMF = -Ldi/dt

So for an inductor or Coil 

(2) EMF=−NdΦB/dt = -Ldi/dt =  Faraday’s Law Applied to an Inductor (coil) 

Self-inductance (L) is the property of an electric circuit (like a coil) that opposes any change in the electric current flowing through it.

It is expressed as the proportionality constant L equating the rate of change of current with time to the change of flux with time.

L, self inductance, 

• has units of volt-seconds/amp =  Vs/A (called a Henry).
• only depends on the geometry and material used and
• just like capacitance and resistance, it does not depend on i or V.

When the switch in a circuit with an inductor is suddenly opened,

• the current attempts to drop instantaneously.
• As the current rapidly drops, the inductor coil, obeying Lenz’s Law,  opposes this with an increase in EMF with enough energy to jump the air gap in the open switch (you get a spark).
• The spark is a result of the inductor’s stored energy rapidly discharging across the gap in a final attempt to keep the current flowing

Self Inductance in a Solenoid

The inducting coil we described above is called a solenoid.

It is a device composed of a helix (a coil of wire) that, when an electric current passes through it, generates a uniform magnetic field inside the coil.

We used the expression NΦ_B = Li to derive  EMF=−NdΦ_B/dt = -Ldi/dt.   

Let’s take a closer look at it.

(3) NΦ_B = Li ; Flux Linkage

• where NΦ_B is described as the “flux linkage”. 
• This equation is analogous to the momentum equation: ρ (momentum) = (inertia or mass)(velocity)
• Just like mass is a body’s resistance to a change in its velocity (its inertia),
• a coil’s inductance ,L , is an electrical inertia,  resisting changes in the current through it.
• Momentum is a measure of the “quantity of motion” stored in an object while the
• flux linkage (NΦ_B) is a measure of the magnetic “inertia” stored in the inductor’s magnetic field.

Just as a moving object carries momentum proportional to its mass and velocity, an inductor carries a magnetic quantity (flux linkage) proportional to its inductance and the current flowing through it.

Rearranging (3), we get   

(4) L = NΦ_B/i

We have an expression for the magnetic flux:

(5) ΦB = Magnetic Flux = (B)(Ap) =BAcos(θ) = B̄•Ā for a Uniform B and Flat Surface

Picture: Single Coil of a Solenoid 

In the case of a solenoid coil, θ = 0 . cos(0) = 1, so,

(6) ΦB = BA

Substituting (6) into (4) we get:

(7) L = NBA/i

We want find an expression for B and substitute it into (7).

Let’s use the expression relating i to B called Ampère’s Circuital Law:

(8) ∮B̄•dL̄ = μ0ienc ; Ampere’s Circuital Law  

It states that the Path integral of B̄⋅dL̄ on a closed curve (the Amperian Loop) is equal to μ₀ times the net enclosed current passing through a surface defined by the curve.

• ∮ is a closed line integral.
  • i.e. summing up the contributions of the magnetic field along a complete, closed path, known as an Amperian loop.
  • The integral path starts and ends at the same point.
• B̄ = Magnetic Field
  • This is the magnetic field vector.
  • B represents the magnitude and direction of the magnetic field at every point along the Amperian loop.
  • Its SI unit is the tesla (T).
• dL̄ = Infinitesimal Line Element
  • This is a small, infinitesimal vector segment of the closed loop.
  • It points in the direction of the integration along the loop.
  • Its SI unit is the meter (m).
• The dot product B̄•dL̄ calculates the component of the magnetic field that is parallel to the path element.
• ∮B̄•dL̄ ;  This term is called the circulation of the magnetic field.

• μ₀ = Permeability of Free Space.

  • A constant that represents the ability of a vacuum to support the formation of a magnetic field.
  • Its value is approximately 4π×10 ⁻⁷ T⋅m/A.
• i_enc = Enclosed Current:
  • This is the net electric current that passes through the surface defined by the closed Amperian loop.
  • It is a scalar quantity, and its SI unit is the ampere (A).

We can simplify the left hand side of (8) by assuming we have a constant B field in the middle of the coil.

We can draw a hypothetical closed Amperian Loop as shown in the picture below. 

Picture: Solenoid Inductor With Amperian Loop

The only component of ∮B̄•dL̄ that is non zero in the Amperian Loop is the side of the rectangle aligned with the constant (straight) B field. 

So equation (8) simplifies to

(9) Bx = μ0ienc

ienc can be expressed as Nenci :

(10) Bx = μ0Nenci

But Nenc = N(x/ds). Substituting for Nenc into (10) gives us

Bx = μ0N(x/ds)i which simplifies to

(11) B = μ0Ni/ds ; Magnetic Field Magnitude in a Solenoid

Substitute (11) into equation (7) to get

(12) L = N²μ0A/ds ; Self Inductance of a Solenoid (no core)

where 

• L = Self Inductance
• N = number of loops in the coil
• μ0= a constant; the Permeability of Free Space (for air or vacuum)
• A = bounded area of each loop

• ds = length of the solenoid coil

If we wrap the coil around a ferromagnetic rod or core, we can achieve a much higher inductance.

We can modify equation (12) to reflect this by defining a total permeability term: 

• The total permeability (μ) of any material is defined as the product of the permeability of free space and the relative permeability of that specific material
• μ = μrμs
• For air/vacuum: μr ≈1
• For ferromagnetic cores: μr can be very large (e.g., 5,000 for soft iron).

So Equation (12) becomes, 

(13) L = N²μrμ0A/ds ; Self Inductance of a Solenoid (with core)

• μr = a constant; the Permeability of the core (ferromagnetic material). 
• μ0 = a constant; the Permeability of Free Space (for air or vacuum)

Energy Stored in an Inductor

If we take our simple circuit and bypass the battery we’ll effectively have a short circuit.

Picture: Circuit with Inductor: Short Circuit

The bulb will eventually go out but it will do it progressively. 

This happens because inductors will resist changes in current. 

Since di/dt is decreasing, then the induced EMF is increasing, meaning the resultant induced voltage will induce the current to continue.

So the inductor is actually providing energy to the bulb in this scenario. 

Energy balance around the bulb

Let’s do an energy balance around the load (the bulb).

• We will develop the balance in terms of energy per time, or Power. 
• From Ohm’s Law and Joule’s Law we know that power = (current)(voltage) = P = IV = iV
• Total Energy generated in the Inductor = Energy Transferred to Bulb
• UL =Total energy per unit time =  VLi = (-Ldi/dt)i ;  where units of measure are energy/second (units of power):
• = (ampere)(joules)/[(ampere)(second)]
• = (ampere)(joules)/(coulomb)
• =  (current)(voltage) = iVL

If the power was constant we could just multiply by the total time to get the total energy, but power is not constant. 

This is why we integrate.

Integration, at its core, is a process of summing an infinite number of infinitesimal parts.

It allows us to calculate a total value for a property that is changing at every single point.

dUL/dt =  (-Ldi/dt)i

dUL =  (-L)(i)(di)

UL = ∫(-L)(i)(di) = -Li²/2 evaluated from i = I to 0  (note: we integrated using the power rule where ∫idi = i²/2)

(14) UL =  LI²/2 ; magnetic potential energy stored in an inductor

An inductor (like a coil of wire) stores energy in the magnetic field created by the current flowing through it, and this formula calculates that stored energy.

• UL = Energy (Magnetic Potential Energy) in Joules (J) = the total amount of energy stored in the inductor’s magnetic field.
• L = Inductance in Henrys (H) = A measure of an inductor’s ability to resist changes in electric current by inducing an opposing voltage (back-EMF).
  • It’s a property determined by the coil’s physical characteristics (number of turns, area, length, and core material).
• $I\; =\; i\; =$Current in Amperes (A) = the instantaneous current flowing through the inductor.

Similarities to Energy in a Capacitor

In the section on capacitance we showed that the potential energy stored in a capacitor is

UC = CV²/2 ; Energy Stored in a Capacitor (in terms of V, C)

which is analogous to the Magnetic potential energy stored in an inductor 

• EMF2; EMFs :The Induced Voltage in the secondary coil. This is the energy transferred from the primary to the secondary circuit.

• di$1$/dt; di$p$/dt : Rate of change of current in the primary coil.
  • Since AC (alternating) current is constantly changing, this term is non-zero.
  • The faster the primary current changes, the larger the induced EMF will be.
• $M$ = The Coupling Coefficient = Mutual Inductance 
  • This is a constant of proportionality that describes the magnetic coupling between the two coils.
  • It tells you how effective the primary coil’s magnetic field is at linking with the secondary coil.

  • $M$ depends on:

    • The number of turns in each coil (N1 and N2).
    • The size and shape of the coils.
    • The distance and orientation between the coils (proximity).
    • The magnetic material (like an iron core, or just air) linking the coils.

• The Negative Sign (Lenz’s Law)
  • The induced EMF ($\text{EMF2 }$$\text{or EMFs}$) will always create a current that produces a magnetic field opposing the change in flux that caused it (di$1$/dt or di$p$/dt)

Finally,

Please note that we are only talking about induced EMF in the equations above.

But the the first coil in our setup also has an EMF voltage provided by the Alternating Current power supply. 

If you were to do a voltage balance in this circuit , you would also have to include the voltage provided by the power source which can be expressed using Ohms Law (V = IR).

Mutual Inductance Expressed Via Self Inductances

Consider our two coil model from the previous section. 

We know from equation (4) that the self inductance for each coil will be

(4a) L₁ = N₁Φ_B1/i₁

(4b) L₂ = N₂Φ_B2/i₂

And from equation (18)

(18a) M$12$ = N$1$ΦB12/i$2$

(18b) M$21$ = N$2$ΦB21/i$1$

Where ,you will recall, the two subscripts a and b in ΦBab mean

• a is the destination, where the magnetic flux is being linked to or being cut over to
• b is the magnetic flux source

We also know that M = M$12$= M$21$

so we can equate equations 18a and 18b.  

(23) M = N$1$ΦB12/i$2$= N$2$ΦB21/i$1$

In reality, some magnetic flux “leaks” and doesn’t reach the other coil.

We account for this with the coupling coefficient (k), which ranges from 0 to 1.

(24) ΦB12  = kΦB2

(25) ΦB21 = kΦB1

Substitute equations 24 and 25 into equation 23.

(26) M = N$1$kΦB2/i$2$ = N$2$kΦB1/i$1$

Express equation 26 as the square of M.

(27) M x M = M2 = (N$2$kΦB2/i$2$)( N$1$kΦB1/i$1$ )  

Express equation 27 in terms of the self inductances via equations 4a and 4b. 

(28) M x M = M2 = (kL₂)( kL₁ )

(29) M = k√(L₂L₁) ; Mutual Inductance (M) between two coils

Equation 29 defines the Mutual Inductance (M) between two coils based on their

• individual self-inductances (L1 and L2) and
• how well they are magnetically linked.

The Coupling Coefficient (k): This is a value between 0 and 1 that represents the efficiency of the magnetic connection.

• If k=1, the coils are “perfectly coupled” (all magnetic flux from one hits the other);
• if k=0, they are completely isolated.

The term √(L₂L₁):  represents the theoretical maximum mutual inductance possible for those two specific coils.

Practical Meaning: In the real world, M is always less than or equal to this geometric mean because some magnetic flux “leaks” into the surrounding air rather than passing through the second coil.

Watch these excellent Khanacademy Videos