Download Inductance and Energy Storage in Magnetic Fields - Prof. Juyang Huang and more Study notes Physics in PDF only on Docsity! Chapter 32 Inductance Ch. 32.1 Self-Inductance • As the resistance changes, the current does not change immediately to its final value I = /R. • • As the current increases or decreases with time, the magnetic flux through the coil due to its current also changes. Which induces an emf that opposes the change. (called self-induction, self-induced emf L). • The current in a circuit can only have graduate change. • Use the Faraday's Law, L = N d B dt = L dI dt L is called the inductance of the coil, that depends on the geometry and material characteristics. Unit: henry (H), 1 H = 1 V•s/A The inductance of a coil: L = N B I , L = L dI / dt Inductance is a measure of the opposition to any change in current. L L iL iL Example: Find the inductance of a solenoid having N turns and length l. Solution: Magnetic field inside the solenoid: B = μ0nI = μ0 N l I The magnetic flux: B = BA = μ0 NA l I A is the cross-section area. The inductance of the solenoid: L = N B I = μ0N 2A l Note: L = μ0 N 2 l2 Al = μ0n 2(volume), L depends on geometry, and n2 32.2 RL Circuits • The switch is closed at t = 0. The current begins to increase. • The inductor produces a back emf that opposes the increasing I. • The inductor acts like a battery whose polarity is opposite that of the real battery, with L = L dI dt . L R L Apply Kirchhoff's loop rule: IR L dI dt = 0 The solution (see textbook): I(t) = R (1 e t / ), is the time constant of the RL circuit: = L/R. Unit: s (a) is the time it takes the current to reach (1-e-1) = 0.632 of its final value, /R Q: How to increase or decrease the time constant of an RL circuit? Example: An RL circuit, the switch is closed at t = 0. (a) Find the time constant: = L/R = 30 10-3 H / 6 = 0.005 s = 5 ms (b) Find the maximum current: I = /R = 12V / 6 = 2 A (c) The time takes for the current to reach 90% of its final value. (1 e t / ) = 0.9, t / = 2.30, t = 11.5 ms • t < 0, the switch is at a, a steady current I = /R is maintained. • At t = 0, switch a b. 6 30mH 12V • A back emf is induced to oppose the sharp decreasing of magnetic flux. • The back emf: L = L dI dt Kirchhoff's rule: IR+ L dI dt = 0 The solution: I(t) = R e t / = I0e t / 32.3 Energy in a Magnetic Field An inductor can store energy in its magnetic field. The back emf by an inductor: L = L dI dt The rate at which energy is stored: I L = IL dI dt = dUB dt or dUB = ILdI The total energy stored in an inductor: dUB0 UB = ILdI 0 I UB = 1 2 LI 2 UB is the energy stored in the magnetic field of an inductor when the current is I. I t I0 0.368I0 I0 = R
