Skip to content Study Mondo Free study resources for students from Grade 4 through AP and test prep. 24 courses, 700+ topics.
Courses Features Company Stay Ahead in School Free weekly study tips, practice sets, and exam strategies. Join 10,000+ students.
ยฉ 2026 Study Mondo. Built for students.
APยฎ is a trademark registered by the College Board, which is not affiliated with, and does not endorse, this website.
Inductance and RL Circuits | Study Mondo
Topics / Electromagnetic Induction / Inductance and RL Circuits Inductance and RL Circuits Self-inductance, mutual inductance, and RL circuit dynamics
๐ฏ โญ INTERACTIVE LESSON
Try the Interactive Version! Learn step-by-step with practice exercises built right in.
Start Interactive Lesson โ Inductance and RL Circuits
Self-Inductance
Changing current in coil induces EMF in same coil:
E = โ L d I d t \mathcal{E} = -L\frac{dI}{dt} E = โ L d t d I โ
where
๐ Practice ProblemsNo example problems available yet.
Explain using: ๐ Simple words ๐ Analogy ๐จ Visual desc. ๐ Example ๐ก Explain
๐งช Practice Lab Interactive practice problems for Inductance and RL Circuits
โพ ๐ Related Topics in Electromagnetic Inductionโ Frequently Asked QuestionsWhat is Inductance and RL Circuits?โพ Self-inductance, mutual inductance, and RL circuit dynamics
How can I study Inductance and RL Circuits effectively?โพ Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Regular review and active practice are key to retention.
Is this Inductance and RL Circuits study guide free?โพ Yes โ all study notes, flashcards, and practice problems for Inductance and RL Circuits on Study Mondo are 100% free. No account is needed to access the content.
What course covers Inductance and RL Circuits?โพ Inductance and RL Circuits is part of the AP Physics C: Electricity & Magnetism course on Study Mondo, specifically in the Electromagnetic Induction section. You can explore the full course for more related topics and practice resources.
๐ก Study Tipsโ Work through examples step-by-step โ Practice with flashcards daily โ Review common mistakes L L L
inductance
Units: 1 henry (H) = 1 Wb/A = 1 Vยทs/A
Inductance of Solenoid Long solenoid: n n n A A A l l l
L = ฮผ 0 n 2 A l = ฮผ 0 N 2 l A L = \mu_0 n^2 Al = \mu_0 \frac{N^2}{l}A L = ฮผ 0 โ n 2 A l = ฮผ 0 โ l N 2 โ A
Mutual Inductance Current I 1 I_1 I 1 โ
E 2 = โ M d I 1 d t \mathcal{E}_2 = -M\frac{dI_1}{dt} E 2 โ = โ M d t d I 1 โ โ
Mutual inductance:
M = ฮฆ 21 I 1 = ฮฆ 12 I 2 M = \frac{\Phi_{21}}{I_1} = \frac{\Phi_{12}}{I_2} M = I 1 โ ฮฆ 21 โ โ = I 2 โ ฮฆ 12 โ โ
(Same value both ways: M 12 = M 21 M_{12} = M_{21} M 12 โ = M 21 โ
Energy Stored in Inductor U L = 1 2 L I 2 U_L = \frac{1}{2}LI^2 U L โ = 2 1 โ L I 2
Energy density in magnetic field:
u B = B 2 2 ฮผ 0 u_B = \frac{B^2}{2\mu_0} u B โ = 2 ฮผ 0 โ B 2 โ
For solenoid:
U = u B โ
volume = B 2 2 ฮผ 0 A l = 1 2 L I 2 U = u_B \cdot \text{volume} = \frac{B^2}{2\mu_0}Al = \frac{1}{2}LI^2 U = u B โ โ
volume = 2 ฮผ 0 โ B 2 โ A l = 2 1 โ L I 2
(Using B = ฮผ 0 n I B = \mu_0 nI B = ฮผ 0 โ n I L = ฮผ 0 n 2 A l L = \mu_0 n^2 Al L = ฮผ 0 โ n 2 A l
RL Circuit: Current Growth Circuit: battery (E \mathcal{E} E R R R L L L
Kirchhoff's loop rule:
E โ I R โ L d I d t = 0 \mathcal{E} - IR - L\frac{dI}{dt} = 0 E โ I R โ L d t d I โ = 0
Differential equation:
d I d t = E L โ R L I \frac{dI}{dt} = \frac{\mathcal{E}}{L} - \frac{R}{L}I d t d I โ = L E โ โ L R โ I
Solution:
I ( t ) = E R ( 1 โ e โ R t / L ) I(t) = \frac{\mathcal{E}}{R}(1 - e^{-Rt/L}) I ( t ) = R E โ ( 1 โ e โ Rt / L )
Time constant:
ฯ L = L R \tau_L = \frac{L}{R} ฯ L โ = R L โ
After time ฯ L \tau_L ฯ L โ ( 1 โ 1 / e ) โ 63 % (1 - 1/e) \approx 63\% ( 1 โ 1/ e ) โ 63%
RL Circuit: Current Decay Remove battery, current decays:
L d I d t + I R = 0 L\frac{dI}{dt} + IR = 0 L d t d I โ + I R = 0
Solution:
I ( t ) = I 0 e โ R t / L I(t) = I_0 e^{-Rt/L} I ( t ) = I 0 โ e โ Rt / L
Current decays exponentially with time constant ฯ L = L / R \tau_L = L/R ฯ L โ = L / R
Energy Considerations Energy from battery: W = โซ 0 โ E I โ d t = E 2 L R 2 W = \int_0^\infty \mathcal{E}I \, dt = \frac{\mathcal{E}^2L}{R^2} W = โซ 0 โ โ E I d t = R 2 E 2 L โ
Energy stored in inductor: U L = 1 2 L ( E R ) 2 U_L = \frac{1}{2}L\left(\frac{\mathcal{E}}{R}\right)^2 U L โ = 2 1 โ L ( R E โ ) 2
Energy dissipated in resistor: U R = 1 2 L ( E R ) 2 U_R = \frac{1}{2}L\left(\frac{\mathcal{E}}{R}\right)^2 U R โ = 2 1 โ L ( R E โ ) 2
(Equal amounts stored and dissipated)
LC Circuit Inductor and capacitor (no resistance):
L d 2 Q d t 2 + Q C = 0 L\frac{d^2Q}{dt^2} + \frac{Q}{C} = 0 L d t 2 d 2 Q โ + C Q โ = 0
Oscillation:
Q ( t ) = Q 0 cos โก ( ฯ t + ฯ ) Q(t) = Q_0\cos(\omega t + \phi) Q ( t ) = Q 0 โ cos ( ฯ t + ฯ )
where ฯ = 1 / L C \omega = 1/\sqrt{LC} ฯ = 1/ L C โ
Energy oscillates between:
Electric: U E = Q 2 / ( 2 C ) U_E = Q^2/(2C) U E โ = Q 2 / ( 2 C )
Magnetic: U B = L I 2 / 2 U_B = LI^2/2 U B โ = L I 2 /2
Total: U = U E + U B U = U_E + U_B U = U E โ + U B โ
LRC Circuit With resistance, oscillations damped:
L d 2 Q d t 2 + R d Q d t + Q C = 0 L\frac{d^2Q}{dt^2} + R\frac{dQ}{dt} + \frac{Q}{C} = 0 L d t 2 d 2 Q โ + R d t d Q โ + C Q โ = 0
Damped oscillation (for R < 2 L / C R < 2\sqrt{L/C} R < 2 L / C โ Q ( t ) = Q 0 e โ R t / ( 2 L ) cos โก ( ฯ โฒ t ) Q(t) = Q_0e^{-Rt/(2L)}\cos(\omega' t) Q ( t ) = Q 0 โ e โ Rt / ( 2 L ) cos ( ฯ โฒ t )
where ฯ โฒ = 1 L C โ R 2 4 L 2 \omega' = \sqrt{\frac{1}{LC} - \frac{R^2}{4L^2}} ฯ โฒ = L C 1 โ โ 4 L 2 R 2 โ โ
Quality factor:
Q = ฯ 0 L R = 1 R L C Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}} Q = R ฯ 0 โ L โ = R 1 โ C L โ
โ