Physics Lesson 16.12.3 - RL Circuits

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

★★★★★ [ 1 Votes ]

Welcome to our Physics lesson on RL Circuits, this is the third lesson of our suite of physics lessons covering the topic of RL Circuits, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

RL Circuits

The rise or fall in the number of charges in a capacitor charging through a resistor results in a rise or fall in the current in the circuit as

I(t) = ∆Q/∆t

For very small time intervals dt, we can write

I(t) = dQ/dt

Similarly, a rise or a fall of the current occurs when a source by an electromotive force ε supplies a single loop circuit containing a resistor R and an inductor L.

Physics Tutorials: This image provides visual information for the physics tutorial RL Circuits

When the switch S is moved at position a, the current in the resistor starts to rise. In absence of inductor, this rise of current from zero to a steady value would be immediate and we used the Ohm's law to find this rise in current, i.e.

I = ε/R

When an inductor is present in the circuit however, a self-induced emf ε_L appears in the circuit. The current generated because of this self-induced emf is in the opposite direction of the current produced by the battery. As a result, it opposes the rise in current (from the Lentz law) and this causes a delay in the rise of current in the circuit. In other words, the current in the circuit is related to the difference of the two emf's: one is the (steady) emf of battery ε and the other is the (changeable) emf self-induced ε_L in the inductor. This last one, has the formula

ε_L = -L di/dt

where i is the current induced in the inductor and L is its inductivity.

Over time, the rise in current due to the self-induced emf in the inductor becomes less rapid. As a result, the current in the circuit approaches the value of the steady current ε calculated through the Ohm's law. However, in presence of an inductor in the circuit, the current never reaches the ε/R value (or using the language of mathematics, we say the current reaches this limit value in an infinite time interval). Therefore, we say:

"An inductor initially opposes the rise in the current in the circuit but after a long time, it acts as a simple conducting wire."

Thus, when the switch is at position a as discussed earlier, the original circuit behaves like the simplified one shown below:

Since numerically the value of the self-induced emf in the best case can equal the value of emf produced by the battery, the current I in the circuit is in the direction of the red arrow (here clockwise). Therefore, the potential in the resistor decreases in the clockwise direction and as a result, the potential difference across the resistor R is

∆V = -i ∙ R

Likewise, since the self-induced emf in the inductor is in the opposite direction to the emf of battery, we write again

ε_L = -L di/dt

Hence, since the emf produced by the battery is clockwise, we write from the Kirchhoff's Second Law (the voltage law), which is based on the law of conservation of energy:

ε + ∆V + ε_i = 0

ε-i ∙ R-L ∙ di/dt = 0

We can rearrange the last equation to isolate ε:

ε = i ∙ R + L ∙ di/dt

The solution of this differential equation in terms of the current I, (using the differentiation techniques which you can find in the math section of this webpage), is

i(t) = ε/R ∙ (1 - e^{- R ∙ t/L})

The form of this equation is similar to that of potential difference (and charge) in a RC circuit.

If we write the term L/R as τ_L (we call it "the inductive time constant), the above equation is written as

i(t) = ε/R ∙ (1 - e^-t/τ_L)

This equation is used to calculate the current at any instant when the current in the circuit is rising. When the current drops, we use the equation

i(t) = ε/R ∙ e^-t/τ_L

to calculate the current in the circuit at any instant t.

Example 1

A 20 Ω resistor is connected to a 12V battery. A 16 cm long inductor having 4000 turns and the area of each loop equal to 8 cm₂, is connected in series to the resistor, as shown in the figure.

Calculate:

The current in the circuit when the switch S is in the actual position (OFF)
The current in the circuit immediately after the switch is closed (ON)
The current in the circuit 0.003 s after the switch is closed (ON)
The current in the circuit 10 s after the switch is closed (ON)
Can you find the exact time when the current becomes steady? Why?
What is the potential difference in the resistor and inductor at t = 0.001 s?

Solution 1

The circuit is initially open (OFF). Hence, no current is flowing through the circuit when the switch is in this position.
We have t = 0 immediately after the switch turns ON. In this case, we use the equation
i(t) = ε/R ∙ (1 - e^{- R ∙ t/L})
to find the current in the circuit at this instant. Thus, we obtain
i(t) = ε/R ∙ (1 - e^{- R ∙ t/L} )
= 12/20 ∙ (1- e ^{- 20 ∙ 0/L} )
= 12/20 ∙ (1 - e⁰ )
= 12/20 ∙ (1 - 1)
= 12/20 ∙ 0
= 0
Therefore, the current is still zero immediately after the switch turns ON, as the self-induced emf in the coil prevents the current from flowing in the circuit.
Before calculating the current in other instants, we must find the self-inductance L in the inductor using the equation
L = μ₀ ∙ N² ∙ A/I
where
N = 4000 = 4 × 103 turns
A = 8 cm₂ = 8 × 10^-4 m²
l = 16 cm = 0.16 m
(μ₀ = 4π × 10^-7 N/A²)
Thus, we obtain for the self-inductance L of the inductor:
L = (4 ∙ 3.14 ∙ 10^-7 N/A² ) ∙ (4 × 10³ )² ∙ (8 × 10^-4 m² )/(0.16 m)
= 10048 × 10^-5 H
= 0.1 H
The current flowing in the circuit at t = 0.003 s is:
i(t) = ε/R ∙ (1 - e^{- R ∙ t/L} )
= 12/20 ∙ (1 - e( ^{- 20 ∙ 0.001/0.1})
= 0.6 A ∙ (1 - e^-0.2 )
= 0.6 ∙ (1 - 0.819)
= 0.6 A ∙ 0.181
= 0.109 A
This means the current is rising but it has not reached the maximum value (0.6 A) yet.
Now, we find the current in the circuit for t = 10s. We have
i(t) = ε/R ∙ (1 - e^{- R ∙ t/L} )
= 12/20 ∙ (1 - e(^{- 20 ∙ 10/0.1} )
= 0.6 ∙ (1 - e^-2000 )
= 0.6 ∙ (1-0)
= 0.6 ∙ 1
= 0.6 A
This results means the current in the circuit is already steady.
It is not possible to find an exact time in which the current reaches the maximum value (0.6 A) and becomes steady because the current rises in an asymptotic fashion, i.e. initially the current rises faster and then the rise slows down more and more, until it becomes undetectable.
We use the value found for the current in (c) to calculate the potential difference in the resistor and inductor in the instant t = 0.001 s. Thus,
∆V_(resistor) = i(t) ∙ R
= (0.109 A) ∙ (20 Ω)
= 2.18 V
and
∆V_(inductor) = ε - ∆V_(resistor)
= 12 V - 2.18 V
= 9.82 V

You have reached the end of Physics lesson 16.12.3 RL Circuits. There are 5 lessons in this physics tutorial covering RL Circuits, you can access all the lessons from this tutorial below.

More RL Circuits Lessons and Learning Resources

Magnetism Learning Material
Tutorial ID	Physics Tutorial Title	Revision Notes	Revision Questions
16.12	RL Circuits
Lesson ID	Physics Lesson Title	Lesson	Video Lesson
16.12.1	What Are RL Circuits? An Overview
16.12.2	A Short Recap on RC Circuits as a Useful Step to Understand the RL Circuits
16.12.3	RL Circuits
16.12.4	What is the Dimension of τ?
16.12.5	What happens to the Current when the Source is removed from the Circuit

Whats next?

Enjoy the "RL Circuits" physics lesson? People who liked the "RL Circuits lesson found the following resources useful:

Knowledge Feedback. Helps other - Leave a rating for this knowledge (see below)
Magnetism Physics tutorial: RL Circuits. Read the RL Circuits physics tutorial and build your physics knowledge of Magnetism
Magnetism Revision Notes: RL Circuits. Print the notes so you can revise the key points covered in the physics tutorial for RL Circuits
Magnetism Practice Questions: RL Circuits. Test and improve your knowledge of RL Circuits with example questins and answers
Check your calculations for Magnetism questions with our excellent Magnetism calculators which contain full equations and calculations clearly displayed line by line. See the Magnetism Calculators by iCalculator™ below.
Continuing learning magnetism - read our next physics tutorial: Energy Stored in a Magnetic Field. Energy Density of a Magnetic Field. Mutual Induction

Help others Learning Physics just like you