Inductance. Self-inductance презентация

Physics 1 Voronkov Vladimir Vasilyevich

Lecture 14 Inductance Self-inductance RL Circuits Energy in a Magnetic Field Mutual inductance LC circuit – harmonic oscillations RLC circuit – damped harmonic oscillations

Self-inductance When the switch is thrown to its closed position, the current does not immediately jump from zero to its maximum value /R. As the current increases with time, the magnetic flux through the circuit loop due to this current also increases with time. This increasing flux creates an induced emf in the circuit. The direction of the induced emf is such that it would cause an induced current in the loop), which would establish a magnetic field opposing the change in the original magnetic field. Thus, the direction of the induced emf is opposite the direction of the emf of the battery; this results in a gradual rather than instantaneous increase in the current to its final equilibrium value. Because of the direction of the induced emf, it is also called a back emf. This effect is called self-induction because the changing flux through the circuit and the resultant induced emf arise from the circuit itself. The emf L set up in this case is called a self-induced emf.

(a) A current in the coil produces a magnetic field directed to the left. (a) A current in the coil produces a magnetic field directed to the left.

Self-induced emf

From last expression it follows that So inductance is a measure of the opposition to a change in current.

Ideal Solenoid Inductance Combining the last expression with Faraday’s law, L = -N dB/dt, we see that the inductance of a closely spaced coil of N turns (a toroid or an ideal solenoid) carrying a current I and containing N turns is

Series RL Circuit An inductor in a circuit opposes changes in the current in that circuit:

Change of variables: x = (/R) – I Change of variables: x = (/R) – I dx = - d where x0 is the value of x at t = 0.

Taking the antilogarithm of the last result: Taking the antilogarithm of the last result: Because I = 0 at t = 0, we note from the definition of x that x0 = /R. Hence, this last expression is equivalent to

The time constant  is the time interval required for I to reach 0.632 (1-e-1) of its maximum value. The time constant  is the time interval required for I to reach 0.632 (1-e-1) of its maximum value.

Energy in an Inductor Multiplying by I the expression for RL–circuit we obtain: So here I is the power output of the battery, I2R is the power dissipated on the resistor, then LIdI/dt is the power delivering to the inductor. Let’s U denote as the energy stored in the inductor, then:

After integration of the last formula: After integration of the last formula: L is the inductance of the inductor, I is the current in the inductor, U is the energy stored in the magnetic field of the inductor.

Magnetic Field Energy Density Inductance for solenoid is: The magnetic field of a solenoid is: Then: Al is the volume of the solenoid, then the energy density of the magnetic field is:

uB is the energy density of the magnetic field uB is the energy density of the magnetic field B is the magnetic field vector 0 is the free space permeability for the magnetic field, a constant. Though this formula was obtained for solenoid, it’s valid for any region of space where a magnetic field exists.

Mutual Inductance A cross-sectional view of two adjacent coils. The current I1 in coil 1, which has N1 turns, creates a magnetic field. Some of the magnetic field lines pass through coil 2, which has N2 turns. The magnetic flux caused by the current in coil 1 and passing through coil 2 is represented by F12. The mutual inductance M12 of coil 2 with respect to coil 1 is:

Mutual inductance depends on the geometry of both circuits and on their orientation with respect to each other. As the circuit separation distance increases, the mutual inductance decreases because the flux linking the circuits decreases.

The emf induced by coil 1 in coil 2 is: The emf induced by coil 1 in coil 2 is: The preceding discussion can be repeated to show that there is a mutual inductance M21. The emf induced by coil 1 in coil 2 is: In mutual induction, the emf induced in one coil is always proportional to the rate at which the current in the other coil is changing.

Although the proportionality constants M12 and M21 have been obtained separately, it can be shown that they are equal. Thus, with M12 = M21 = M, the expressions for induced emf takes the form: Although the proportionality constants M12 and M21 have been obtained separately, it can be shown that they are equal. Thus, with M12 = M21 = M, the expressions for induced emf takes the form: These two expression are similar to that for the self-induced emf:  = - L(dI/dt). The unit of mutual inductance is the henry.

LC Circuit Oscillations If the capacitor is initially charged and the switch is then closed, we find that both the current in the circuit and the charge on the capacitor oscillate between maximum positive and negative values. We assume: the resistance of the circuit is zero, then no energy is dissipated, energy is not radiated away from the circuit.

The energy of the LC system is: The energy of the LC system is: U=const as we supposed no energy loss: Using that I=dQ/dt we can write:

The solution for the equation is: The angular frequency of the oscillations depends solely on the inductance and capacitance of the circuit. This is the natural frequency (частота собственных колебаний) of oscillation of the LC circuit.

Then the current is: Then the current is: Choosing the initial conditions: at t = 0, I = 0 and Q = Qmax we determine that =0. Eventually, the charge in the capacitor and the current in the inductor are:

Graph of charge versus time and Graph of current versus time for a resistanceless, nonradiating LC circuit. NOTE: Q and I are 90° out of phase with each other.

Plots of UC versus t and UL versus t for a resistanceless, nonradiating LC circuit. Plots of UC versus t and UL versus t for a resistanceless, nonradiating LC circuit. The sum of the two curves is a constant and equal to the total energy stored in the circuit.

RLC circuit A series RLC circuit. Switch S1 is closed and the capacitor is charged. S1 is then opened and, at t = 0, switch S2 is closed.

Energy is dissipated on the resistor: Energy is dissipated on the resistor: Using the equation for dU/dt in the LC-circuit (slide 2): Using that I=dQ/dt:

The RLC circuit is analogous to the damped harmonic oscillator, where R is damping coefficient. The RLC circuit is analogous to the damped harmonic oscillator, where R is damping coefficient. Here b is damping coefficient. When b=0, we have pure harmonic oscillations. Solution is: RC is the critical resistance: When R<RC oscillations are damped harmonic. When R>RC oscillations are damped unharmonic.

The charge decays in damped harmonic way when The charge decays in damped harmonic way when

Charge decays in damped inharmonic way when R>RC, then the RLC circuit is overdamped. Charge decays in damped inharmonic way when R>RC, then the RLC circuit is overdamped.

Units in Si Inductance L H (henry): 1H=V*s/A Mutual Inductance M H (henry): 1H=V*s/A Energy density u J/m3