For more information, please see full course syllabus of AP Physics C: Electricity & Magnetism

For more information, please see full course syllabus of AP Physics C: Electricity & Magnetism

## Discussion

## Study Guides

## Download Lecture Slides

## Table of Contents

## Transcription

## Related Books

### LC Circuits

- To analyze an LC circuit, use Faraday’s Law. You cannot correctly use Kirchhoff’s Voltage Law (the loop rule) since the magnetic flux in the circuit is changing.
- LC circuits lead to oscillating voltage and current curves, and are often referred to as resonant circuits.
- The resonant frequency of an LC circuit is equal to 1/Sqrt(LC) in radians per second. The frequency in Hertz is found by dividing the resonant frequency by 2*Pi.

### LC Circuits

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Objectives 0:08
- LC Circuits 0:30
- Assume Capacitor is Fully Charged When Circuit is First Turned On
- Interplay of Capacitor and Inductor Creates an Oscillating System
- Charge in LC Circuit 0:57
- Current and Potential in LC Circuits 7:14
- Graphs of LC Circuits 8:27

### AP Physics C: Electricity and Magnetism Online Course

I. Electricity | ||
---|---|---|

Electric Charge & Coulomb's Law | 30:48 | |

Electric Fields | 1:19:22 | |

Gauss's Law | 52:53 | |

Electric Potential & Electric Potential Energy | 1:14:03 | |

Electric Potential Due to Continuous Charge Distributions | 1:01:28 | |

Conductors | 20:35 | |

Capacitors | 41:23 | |

II. Current Electricity | ||

Current & Resistance | 17:59 | |

Circuits I: Series Circuits | 29:08 | |

Circuits II: Parallel Circuits | 39:09 | |

RC Circuits: Steady State | 34:03 | |

RC Circuits: Transient Analysis | 1:01:07 | |

III. Magnetism | ||

Magnets | 8:38 | |

Moving Charges In Magnetic Fields | 29:07 | |

Forces on Current-Carrying Wires | 17:52 | |

Magnetic Fields Due to Current-Carrying Wires | 24:43 | |

The Biot-Savart Law | 21:50 | |

Ampere's Law | 26:31 | |

Magnetic Flux | 7:24 | |

Faraday's Law & Lenz's Law | 1:04:33 | |

IV. Inductance, RL Circuits, and LC Circuits | ||

Inductance | 6:41 | |

RL Circuits | 42:17 | |

LC Circuits | 9:47 | |

V. Maxwell's Equations | ||

Maxwell's Equations | 3:38 | |

VI. Sample AP Exams | ||

1998 AP Practice Exam: Multiple Choice Questions | 32:33 | |

1998 AP Practice Exam: Free Response Questions | 29:55 |

### Transcription: LC Circuits

*Hello, everyone, and welcome back to www.educator.com.*0000

*I'm Dan Fullerton, and in this lesson, we are going to talk about LC circuits.*0003

*Our objectives include applying Faraday’s law to a simple LC series circuit*0008

*to obtain a differential equation for charge as a function of time.*0013

*To solve the differential equation.*0016

*To calculate the current in capacitor voltage as a function of time.*0019

*Finally, sketching graphs of the current through the voltage across the capacitor for the simple LC series circuits.*0022

*Let us take a look at our LC circuit.*0029

*We have an inductor and capacitor and typically when we analyze these,*0032

*what we are going to assume is when the circuit is first turned on, our capacitor is charged and our inductor is not.*0036

*The interplay of the capacitor and the inductor is going to create an oscillating system very similar in fashion to simple harmonic motion,*0043

*basically, looking at making the electrical version of a pendulum.*0051

*Let us see if we can analyze the circuit a little bit.*0056

*As we look at charge in here, let us first define in our circuit the direction for current flow, let us call that our current.*0061

*We have our potential across our capacitor DC ± recognizing that the electric field in here*0067

*is going to go from the positive side to the negative side.*0074

*Over here on our inductor, we have the potential across it which is L DI DT, that is our positive and our negative side.*0079

*Again, it is very important to note that the electric field in here is 0, the electric field inside the conductor.*0089

*With that, we can use Faraday's law to start to analyze our circuit.*0097

*The integral / the closed loop of E dot DL = - the derivative of the magnetic flux with respect to time, which as we look here is just going to be - L DI DT.*0103

*Since, we know that the voltage across our capacitor VC is Q/C, we can rewrite this as –Q/ C as we go this way around our circuit,*0125

*no electric fields and no contribution to the left hand side, over from our inductor is going to be equal to our - L DI DT.*0137

*Or rearranging this a little bit, we could say that Q/ C - L DI DT = 0.*0149

*But Q and our charge is changing and DI DT is also a moving change in charge.*0161

*These are functions of the same variable.*0169

*Let us see if we can put this all in terms of the same variable.*0172

*We are going to do that by recognizing first, that our current I is - DQ DT.*0176

*Negative because we are discharging that.*0184

*DI DT therefore, must be the opposite of the second derivative of Q with respect to T – D² Q/ DT².*0188

*I will rewrite our equation, we have Q/ C - L times the second derivative of Q with respect to time = 0.*0201

*Or in a more standard version of writing this, we can write this as D² Q/ DT² + Q/ LC = 0.*0220

*We have got a function, a differential equation where the second derivative of something + that something gives you 0.*0236

*Only way I know of that you can have an answer to a function like that is if you are looking at something like a cos or sin.*0244

*Remember, the derivative of sin is the cos, derivative of cos is the opposite of the sin.*0251

*The second derivative of something + that something can give you 0, right away sin is a real function.*0256

*Let us also go and let us define ω as 1/ √ LC.*0264

*This is actually ω² Q.*0272

*When we do that, we are going to solve this equation, what we are going to find is*0275

*that the form that fits this equation is some constant A times the cos of ω T, where ω is 1/ √ of LC + B sin ω T.*0280

*Now we can use our boundary conditions to figure out exactly what these A and B are.*0300

*One of the boundary conditions that we know is that the charge at T = 0 on our capacitor is Q0, our initial charge.*0305

*If T = 0, that sin function, sin 0 is 0, this whole term goes away.*0318

*T equal 0 cos of 0 is 1, that means A must be equal to Q0.*0324

*This implies then that A = Q0.*0330

*To look at our second boundary condition, when you recognize that our current I which is DQ DT is going to be,*0335

*as I go through this and take our derivative, A cos ω T, derivative of cos is opposite of the sin.*0346

*I'm going to get - ω A sin ω T from that first term + derivative of sin is cos, we will have ω B cos ω T.*0352

*With the boundary condition that the current at time T = 0 has to equal 0, everything is in that capacitor.*0369

*At T = 0, this whole first term goes away, that means cos 0,1 ω B must equal 0.*0379

*Ω is 1/ √ LC that can equal 0.*0388

*Therefore, B must be 0.*0391

*Putting all of this together, we can write our solution.*0396

*Therefore, Q of T must be A cos ω T, where A is Q0 so that is going to be Q0 cos ω T.*0402

*Since B is 0, this term goes way.*0415

*Here is our solution for the charge as a function of time.*0418

*Notice it varies as a function of time.*0425

*It is going to follow a simple harmonic motion type pattern.*0427

*We know Q of T = Q0 cos ω T, therefore the potential across our capacitor is just Q/ C, so that is going to be Q₀/ C cos ω T.*0435

*If we want to take a look at current as a function of time, I of T is just - DQ DT, which is going to be ω Q₀ sin ω T.*0457

*If we wanted to write this with our ω equal to 1/ √ of LC, oftentimes you will see this piece written as I of T = Q₀/ √ of LC sin ω T.*0476

*Let us graph these so you can really see the sign, we will make sure.*0504

*When I do this, if I look at the current as a function of time or the potential as a function of time for the circuit,*0507

*we get the sign in this pattern where we see the current just ahead of the voltage or depending on how you are looking at it.*0514

*Time you could say voltage ahead of current, just depends on your perspective.*0522

*But we see we have this repeating pattern of current and voltage as we have the electrical signal oscillating inside our simple LC circuit.*0526

*Assuming that it is a completely efficient circuit, it would go on forever and ever.*0536

*Now in reality, you have a dampening effect as energy is always lost in some of these circuits.*0541

*Eventually those appear out to 0.*0547

*But for a short term ideal situation, you have an ongoing harmonic oscillator, an electrical oscillator.*0549

*As we look at the angular frequency, ω = 1/ √ of LC, if we want our frequency in Hertz,*0557

*we can take a look at frequency is ω / 2 π which is going to be 1/ 2 π √ LC.*0565

*Hopefully that gives you a bit of insight into LC circuits as electrical oscillators.*0579

*Thank you so much for watching www.educator.com and make it a great day.*0584

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