Hey, guys. So let's check out this example. We're gonna work this one out together of calculating the self-inductance of a toroidal solenoid. So try and say that 5 times fast. Given some information about the geometry of a toroid, in the first part of the question, we're supposed to figure out what is the self-inductance of a toroid. And then in the second part, we're gonna figure out what the induced EMF is, assuming that the current is changing. Alright. So I want to take this actually really slow because remember that toroids can be pretty complicated shapes. But just remember that they're basically just big donuts of slinkies that have been wrapped up inside of each other. So that's actually pretty bad. Let's see if I can get this right on the 3rd try. There we go. So we've got this doughnut. And remember that a toroid is basically like if you took a slinky with a bunch of coils and you sort of wrapped it around itself. So you have a bunch of coils like this. And I'm gonna try to do this as best as I can, sort of like that. That's pretty bad, but it's going to have to do. Alright? So we have the self-inductance. Let's see. So in part a, if we want to figure out what the self-inductance is, remember that it is L, and we have some information like the number of turns, which is 500, the cross-sectional area, which is 6.25 centimeters squared. So we have that we have n number of turns, which is 500. And then the cross-sectional area represents the area of one of the little loops that it makes of one of the slinky turns. Right? So that's the cross-sectional area. And then the mean radius of 4 centimeters. Well, that's actually remember that the mean radius is not the radius in between the slinky like that. It's not the radius of the slinky itself. It is basically from the center of the donut out to the midpoint of that slinky. That is actually the mean radius. That's little r. And so we know that that little r is equal to 0 points, that's actually equal to 4 centimeters. So first, what I'm gonna do is I'm just gonna convert these. This is actually 6.25 times 10 to the minus 4 meters squared, because we have to apply that conversion from centimeters to meters twice. And then this is just equal to 0.04. Okay, cool. So now that we're done with the diagram and labeling all of our variables and all that stuff, what is the equation that's going to relate the self-inductance with all of these variables about the geometry of the coil? Well, remember that we can sort of relate all of these variables in together using the self-inductance formula, which is the number of turns times the flux divided by the currents. So we know what the number of turns is, and we know that this current, even though if we don't have it in our equation or in our problem, it's going to cancel out because the inductance always basically cancels out that current term in there. So all we have to do is just relate the φ, or just figure out an expression for the magnetic flux, which is BA cos(θ). So just a refresher, what does the magnetic field look like inside of a toroid? Well, it's basically kind of like a solenoid, but it always sort of points along the midline of that slinky. So it always basically goes around the center like that. So that B term would just go around like this. And because we're talking about the area, the area is going to be the area of the cross-sections of one of the loops right here. So that means that always at all times, the area vector because these things point along the same direction always. So that means that the magnetic flux right here, that's φB, is going to be Remember that this is going to be the magnetic field This is going to be the magnetic field of a toroid, and this is just the cross-sectional area which we're just given right here. Well, remember that for a toroid, you might have to look in your notes for this equation. The magnetic field is μ0 n i / 2π r, and then we just have the area right here. Okay? So what I can do is I can basically just now plug this whole expression for the flux back into this equation for the self-inductance. So that means that the self-inductance here is going to be n / i times this whole entire flux formula right here, which is equal to μ0 n ia / 2π r. I know it's kind of, kind of confusing there. There we go. And so we have this n that pops up twice, so it's gonna pick up a squared. And just like we-expected, the current term will go away because it's in the top and the bottom. So that means that the self-inductance here, in terms of actual numbers, is going to be 4π times 10 to the minus 7. That's the μ0 term. Then this n term actually gets multiplied twice because there's 2 of them. So I have 500 squared. Now the cross-sectional area is 6.25 times 10 to the minus 4, right? And then I have divided by, let's see, that's gonna be 2π times let's see, I actually don't need that parentheses. 2π times the mean radius, which is 0.04. And so if you go ahead and work this out in your calculators, you're gonna get a self-inductance of 7.81 times 10 to the minus 4 henries. Alright. So now that's the self-inductance of this toroidal solenoid. So you just relate it to the flux, the current term will cancel out, and it's just sort of like a property of this toroid. Cool. So now in the second part here, now that we know what this self-inductance is, if we have the current that is constantly decreasing, how can we relate this to the induced EMF of the coil? So, basically, what we're being asked for in this second part is what is epsilon of this L right here. So, sometimes you'll see this L for an inductor. What is ε_induced? Okay. So we know that the ΔI, the change in current is gonna be I_final - I_initial, which is going to be 2 amps - 5 amps. So in other words, the change in current was just equal to negative 3 amps. And we had the δ_time. So in other words, the change occurred over 3 milliseconds. So that's 0.003 seconds. Okay. So how do we get the inductance Or sorry. How do we get the self-induced EMF from the change in current over change in time? Well, remember that these variables here are related to the equation ε = -L ΔI/Δt. So let's see. We're trying to figure out what this induced EMF is, so we're trying to figure out what ε is. We know what the self-inductance is because we just calculated that in the last part. And now we have what the change in current over change in time is. So we have all of these variables. So that means that my ε_induced is going to be negative. Now, I have 7.81 times 10 to the minus 4. And now, I have the ΔI/Δt. So in other words, this is gonna be negative 3 divided by 0.003. What we'll see is that the two negative signs will actually cancel, and you should get an answer that is 0.78 volts. And that's our answer for the self-induced EMF. Alright, guys. That wraps this up. Let me know if you have any questions.
Table of contents
- 0. Math Review31m
- 1. Intro to Physics Units1h 23m
- 2. 1D Motion / Kinematics3h 56m
- Vectors, Scalars, & Displacement13m
- Average Velocity32m
- Intro to Acceleration7m
- Position-Time Graphs & Velocity26m
- Conceptual Problems with Position-Time Graphs22m
- Velocity-Time Graphs & Acceleration5m
- Calculating Displacement from Velocity-Time Graphs15m
- Conceptual Problems with Velocity-Time Graphs10m
- Calculating Change in Velocity from Acceleration-Time Graphs10m
- Graphing Position, Velocity, and Acceleration Graphs11m
- Kinematics Equations37m
- Vertical Motion and Free Fall19m
- Catch/Overtake Problems23m
- 3. Vectors2h 43m
- Review of Vectors vs. Scalars1m
- Introduction to Vectors7m
- Adding Vectors Graphically22m
- Vector Composition & Decomposition11m
- Adding Vectors by Components13m
- Trig Review24m
- Unit Vectors15m
- Introduction to Dot Product (Scalar Product)12m
- Calculating Dot Product Using Components12m
- Intro to Cross Product (Vector Product)23m
- Calculating Cross Product Using Components17m
- 4. 2D Kinematics1h 42m
- 5. Projectile Motion3h 6m
- 6. Intro to Forces (Dynamics)3h 22m
- 7. Friction, Inclines, Systems2h 44m
- 8. Centripetal Forces & Gravitation7h 26m
- Uniform Circular Motion7m
- Period and Frequency in Uniform Circular Motion20m
- Centripetal Forces15m
- Vertical Centripetal Forces10m
- Flat Curves9m
- Banked Curves10m
- Newton's Law of Gravity30m
- Gravitational Forces in 2D25m
- Acceleration Due to Gravity13m
- Satellite Motion: Intro5m
- Satellite Motion: Speed & Period35m
- Geosynchronous Orbits15m
- Overview of Kepler's Laws5m
- Kepler's First Law11m
- Kepler's Third Law16m
- Kepler's Third Law for Elliptical Orbits15m
- Gravitational Potential Energy21m
- Gravitational Potential Energy for Systems of Masses17m
- Escape Velocity21m
- Energy of Circular Orbits23m
- Energy of Elliptical Orbits36m
- Black Holes16m
- Gravitational Force Inside the Earth13m
- Mass Distribution with Calculus45m
- 9. Work & Energy1h 59m
- 10. Conservation of Energy2h 54m
- Intro to Energy Types3m
- Gravitational Potential Energy10m
- Intro to Conservation of Energy32m
- Energy with Non-Conservative Forces20m
- Springs & Elastic Potential Energy19m
- Solving Projectile Motion Using Energy13m
- Motion Along Curved Paths4m
- Rollercoaster Problems13m
- Pendulum Problems13m
- Energy in Connected Objects (Systems)24m
- Force & Potential Energy18m
- 11. Momentum & Impulse3h 40m
- Intro to Momentum11m
- Intro to Impulse14m
- Impulse with Variable Forces12m
- Intro to Conservation of Momentum17m
- Push-Away Problems19m
- Types of Collisions4m
- Completely Inelastic Collisions28m
- Adding Mass to a Moving System8m
- Collisions & Motion (Momentum & Energy)26m
- Ballistic Pendulum14m
- Collisions with Springs13m
- Elastic Collisions24m
- How to Identify the Type of Collision9m
- Intro to Center of Mass15m
- 12. Rotational Kinematics2h 59m
- 13. Rotational Inertia & Energy7h 4m
- More Conservation of Energy Problems54m
- Conservation of Energy in Rolling Motion45m
- Parallel Axis Theorem13m
- Intro to Moment of Inertia28m
- Moment of Inertia via Integration18m
- Moment of Inertia of Systems23m
- Moment of Inertia & Mass Distribution10m
- Intro to Rotational Kinetic Energy16m
- Energy of Rolling Motion18m
- Types of Motion & Energy24m
- Conservation of Energy with Rotation35m
- Torque with Kinematic Equations56m
- Rotational Dynamics with Two Motions50m
- Rotational Dynamics of Rolling Motion27m
- 14. Torque & Rotational Dynamics2h 5m
- 15. Rotational Equilibrium3h 39m
- 16. Angular Momentum3h 6m
- Opening/Closing Arms on Rotating Stool18m
- Conservation of Angular Momentum46m
- Angular Momentum & Newton's Second Law10m
- Intro to Angular Collisions15m
- Jumping Into/Out of Moving Disc23m
- Spinning on String of Variable Length20m
- Angular Collisions with Linear Motion8m
- Intro to Angular Momentum15m
- Angular Momentum of a Point Mass21m
- Angular Momentum of Objects in Linear Motion7m
- 17. Periodic Motion2h 9m
- 18. Waves & Sound3h 40m
- Intro to Waves11m
- Velocity of Transverse Waves21m
- Velocity of Longitudinal Waves11m
- Wave Functions31m
- Phase Constant14m
- Average Power of Waves on Strings10m
- Wave Intensity19m
- Sound Intensity13m
- Wave Interference8m
- Superposition of Wave Functions3m
- Standing Waves30m
- Standing Wave Functions14m
- Standing Sound Waves12m
- Beats8m
- The Doppler Effect7m
- 19. Fluid Mechanics2h 27m
- 20. Heat and Temperature3h 7m
- Temperature16m
- Linear Thermal Expansion14m
- Volume Thermal Expansion14m
- Moles and Avogadro's Number14m
- Specific Heat & Temperature Changes12m
- Latent Heat & Phase Changes16m
- Intro to Calorimetry21m
- Calorimetry with Temperature and Phase Changes15m
- Advanced Calorimetry: Equilibrium Temperature with Phase Changes9m
- Phase Diagrams, Triple Points and Critical Points6m
- Heat Transfer44m
- 21. Kinetic Theory of Ideal Gases1h 50m
- 22. The First Law of Thermodynamics1h 26m
- 23. The Second Law of Thermodynamics3h 11m
- 24. Electric Force & Field; Gauss' Law3h 42m
- 25. Electric Potential1h 51m
- 26. Capacitors & Dielectrics2h 2m
- 27. Resistors & DC Circuits3h 8m
- 28. Magnetic Fields and Forces2h 23m
- 29. Sources of Magnetic Field2h 30m
- Magnetic Field Produced by Moving Charges10m
- Magnetic Field Produced by Straight Currents27m
- Magnetic Force Between Parallel Currents12m
- Magnetic Force Between Two Moving Charges9m
- Magnetic Field Produced by Loops and Solenoids42m
- Toroidal Solenoids aka Toroids12m
- Biot-Savart Law (Calculus)18m
- Ampere's Law (Calculus)17m
- 30. Induction and Inductance3h 37m
- 31. Alternating Current2h 37m
- Alternating Voltages and Currents18m
- RMS Current and Voltage9m
- Phasors20m
- Resistors in AC Circuits9m
- Phasors for Resistors7m
- Capacitors in AC Circuits16m
- Phasors for Capacitors8m
- Inductors in AC Circuits13m
- Phasors for Inductors7m
- Impedance in AC Circuits18m
- Series LRC Circuits11m
- Resonance in Series LRC Circuits10m
- Power in AC Circuits5m
- 32. Electromagnetic Waves2h 14m
- 33. Geometric Optics2h 57m
- 34. Wave Optics1h 15m
- 35. Special Relativity2h 10m
30. Induction and Inductance
Self Inductance
Video duration:
7mPlay a video:
Related Videos
Related Practice
Self Inductance practice set
- Problem sets built by lead tutorsExpert video explanations