Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
Welcome to our Physics lesson on Angular Frequency, this is the fourth lesson of our suite of physics lessons covering the topic of Alternating Current. LC Circuits, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
Angular Frequency
We can check whether the equation
Q(t) = Qmax ∙ cos(ω ∙ t + φ)
is a solution for the differential equation
L ∙ d2 Q/dt2 + 1/C ∙ Q = 0
The first derivative of equation
Q(t) = Qmax ∙ cos(ω ∙ t + φ)
is
i(t) = dQ/dt = -ω ∙ Qmax ∙ sin(ω ∙ t + φ)
while the second derivative of the same equation is
d2 Q/dt2 = -ω2 Qmax ∙ cos(ω ∙ t + φ)
Hence, we can write
-L ∙ ω2 ∙ Qmax ∙ cos(ω ∙ t + φ) + 1/C ∙ Qmax ∙ cos(ω ∙ t + φ) = 0
Cancelling the term Qmax ∙ cos(ω ∙ t + φ) from the above equation, we obtain
-L ∙ ω2 + 1/C = 0
Rearranging the last equation to isolate the angular frequency ω, we obtain
1/C = L ∙ ω2
ω2 = 1/L ∙ C
ω = 1/√L ∙ C
This is the same equation for angular frequency obtained earlier through another method, so the approach is correct. The equation of Q(t) is a solution for the differential equation
L ∙ d2 Q/dt2 + 1/C ∙ Q = 0
As for the electrical and magnetic oscillations, we can write for the electrical energy stored in a LC circuit at a given time
We (t) = Q2/2C = Qmax2/2C ∙ cos2 (ω ∙ t + φ)
while for the magnetic energy stored in the magnetic field of inductor at a given time, we have
WM (t) = L ∙ i2/2 = L ∙ ω2 ∙ Qmax2/2 ∙ sin2 (ω ∙ t + φ)
Substituting ω = 1/√L ∙ C, we obtain for the magnetic energy stored in a LC circuit
WM = L ∙ Qmax2/2L ∙ C ∙ sin2 (ω ∙ t + φ)
= Qmax2/2C ∙ sin2 (ω ∙ t + φ)
Example 4
A LC circuit is operating at standard values of mains electricity (240V, 50Hz). Calculate:
- The current flowing in the circuit 1.753 s after the switch is turned ON if the resistance in the circuit is 30Ω
- The energy stored in the magnetic field of inductor at this instant if its inductance is 0.8H
- The maximum charge stored in the capacitor plates
Take the initial phase of oscillations as zero (φ = 0).
Solution 4
Clues:
ΔVmax = 240V
f = 50 Hz
φ = 0
R = 30Ω
L = 0.8H
i(1.753) = ?
Wm (1.753) = ?
Qmax = ?
- The maximum current in the circuit is
imax = ∆Vmax/R
= 240 V/30Ω
= 8A
Thus, the amount of current flowing in the circuit at t = 1.753 s, is i(t) = -imax ∙ sin (ω ∙ t + φ)
i(t) = -imax ∙ sin (2π ∙ f ∙ t + φ)
i(1.753) = -8 ∙ sin (2 ∙ π ∙ 50 ∙ 1.753 + 0)
i(1.753) = -8 ∙ sin (175.3 ∙ π)
i(1.753) = -8 ∙ sin (175.3 ∙ π)
i(1.753) = -8 ∙ sin (174 ∙ π + 1.3 ∙ π)
i(1.753) = -8 ∙ sin (1.3 ∙ π)
i(1.753) = -8 ∙ (-0.80)
= 6.4A
- The energy stored in the inductor at the given instant (t = 1.753 s), is
WM = L ∙ i2/2
= (0.8 H) ∙ (6.4A)2/2
= 16.384 J
- Given that the energy stored in the magnetic field of a LC circuit is
WM (t) = L ∙ i2/2 = L ∙ ω2 ∙ Qmax2/2 ∙ sin2 (ω ∙ t + φ)
we obtain for the maximum electric charge Qmax stored in the capacitor plates: imax2 = ω2 ∙ Qmax2
or Qmax = imax/ω
= imax/2 ∙ π ∙ f
= (8A)/2 ∙ 3.14 ∙ (50 Hz)
= 0.0255 C
You have reached the end of Physics lesson 16.14.4 Angular Frequency. There are 5 lessons in this physics tutorial covering Alternating Current. LC Circuits, you can access all the lessons from this tutorial below.
More Alternating Current. LC Circuits Lessons and Learning Resources
Magnetism Learning Material| Tutorial ID | Physics Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 16.14 | Alternating Current. LC Circuits | | | | |
| Lesson ID | Physics Lesson Title | Lesson | Video
Lesson |
|---|
| 16.14.1 | LC Oscillations | | |
| 16.14.2 | Electrical to Mechanical Analogy between Two Oscillating Systems | | |
| 16.14.3 | LC Oscillations - A Quantitative Approach | | |
| 16.14.4 | Angular Frequency | | |
| 16.14.5 | Potential Difference in a LC Circuit | | |
Whats next?
Enjoy the "Angular Frequency" physics lesson? People who liked the "Alternating Current. LC Circuits lesson found the following resources useful:
- Angular Frequency Feedback. Helps other - Leave a rating for this angular frequency (see below)
- Magnetism Physics tutorial: Alternating Current. LC Circuits. Read the Alternating Current. LC Circuits physics tutorial and build your physics knowledge of Magnetism
- Magnetism Revision Notes: Alternating Current. LC Circuits. Print the notes so you can revise the key points covered in the physics tutorial for Alternating Current. LC Circuits
- Magnetism Practice Questions: Alternating Current. LC Circuits. Test and improve your knowledge of Alternating Current. LC 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: Introduction to RLC Circuits
Help others Learning Physics just like you
Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
We hope you found this Physics lesson "Alternating Current. LC Circuits" useful. If you did it would be great if you could spare the time to rate this physics lesson (simply click on the number of stars that match your assessment of this physics learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of physics and other disciplines.
Magnetism Calculators by iCalculator™