PHYS 012A Lecture Notes - Lecture 23: Rl Circuit, Inductor, Rc Circuit
12 Jun 2018
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Chapter 32: Inductance
Self – Inductance
- “ith losed, urret does’t reah ax alue
- As current increases with time, magnetic flux increases also
- 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.
- The direction of the induced emf is opposite the direction of the emf of the battery.
- This results in a gradual increase in the current to its final equilibrium value.
- This effect is called self-inductance.
o Because the changing flux through the circuit and the resultant induced emf arise from
the circuit itself.
- The ef εL is called a self-induced emf.
Self - Inductance Equations
- An induced emf is always proportional to the time rate of change of the current.
- The emf is proportional to the flux, which is proportional to the field and the field is proportional
to the current.
- L is a constant of proportionality called the inductance of the coil.
o It depends on the geometry of the coil and other physical characteristics.
Inductance of a Coil
- A closely spaced coil of N turns carries current I have an inductance of
Inductance of a Solenoid
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RC Circuit
- Circuit element with a large self-inductance is called an inductor
Effect of an Inductor in a Circuit
- Inductances results in a back emf
- Therefore, the inductor in a circuit opposes changes in current in that circuit.
o The inductor attempts to keep the current the same way it was before the change
occurred.
o The idutor a ause the iruit to e sluggish as it reats to hages i the oltage.
RL Circuit, Analysis
- An RL circuit contains an inductor and a resistor in series
- Assume S2 is i the a positio
- When switch S1 is closed (at time t = 0), the current begins to
increase.
- At the same time, a back emf is induced in the inductor that
opposes the original increasing current.
- If there was no inductor, the exponential term would go to zero, and the current would reach its
max.
- Current initially increases very rapidly then gradually as it approaches equilibrium.
RL Circuit, di/dt vs. Time Graph, Charging
- time rate of change of the current is a max at t=0.
- Falls off exponentially as t approaches infinity
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RL Circuit Without A Battery
Energy in a Magnetic Field
- In a circuit with an inductor, the battery must supply more energy than in a circuit without an
inductor.
- Part of the energy supplied by the battery appears as internal energy in the resistor.
- The remaining energy is stored in the magnetic field of the inductor.
- Kirhoff’s oltage rule tells us:
o Ie is the rate at which energy is being supplied by the battery.
o I2R is the rate at which the energy is being delivered to the resistor.
o Therefore, LI (dI/dt) must be the rate at which the energy is being stored in the
magnetic field of the inductor.
- Let U denote the energy stored in the inductor at any time.
- The rate at which the energy is stored is:
Energy Density of a Magnetic Field
- Given U = ½ L I2 and assume (for simplicity) a solenoid with L = n2 V
- Since V is the volume of the solenoid, the magnetic energy density, uB is
- This applies to any region in which a magnetic field exists (not just the solenoid).
Energy Storage Summary
- A resistor, inductor and capacitor all store energy through different mechanisms.
o Charged capacitor
▪ Stores energy as electric potential energy
o Inductor
▪ When it carries a current, stores energy as magnetic potential energy
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Document Summary
As current increases with time, magnetic flux increases also. 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. The direction of the induced emf is opposite the direction of the emf of the battery. This results in a gradual increase in the current to its final equilibrium value. This effect is called self-inductance: because the changing flux through the circuit and the resultant induced emf arise from the circuit itself. The e(cid:373)f l is called a self-induced emf. An induced emf is always proportional to the time rate of change of the current. The emf is proportional to the flux, which is proportional to the field and the field is proportional to the current. L is a constant of proportionality called the inductance of the coil.
